Treatise on differential equations. Supplementary volume

TREATISE

DIFFERENTIAL EQUATIONS.

SUPPLEMENTAL Y VOL UME.

N

PREFACE.

THE present volume contains all that Professor Boole
•wrote for the purpose of enlarging his Treatise on Differential
Equations. Had he lived to publish the second edition he
would doubtless have incorporated his more recent investi-
gations with the original work, and it is therefore necessary
to explain why another plan has been adopted.

In some cases Professor Boole had indicated that certain
portions of the original work were to be omitted and their
places supplied from the manuscripts; but on examination
it appeared that in subsequent passages of the work there
were references and allusions to the portions thus marked to
be omitted which would not apply to the substituted matter.
Thus in attempting to carry out the directions it would
have been necessary to accept the responsibility of making
many alterations, and consequently to incur the risk of fail-
ing in the attempt to improve the original form.

Moreover the Treatise had been for some time out of
print, and the long delay which must have been caused by
the labour of reconstruction would have produced serious
inconvenience to students at Cambridge and elsewhere. Pro-
fessor Boole himself was always especially anxious to consult

VI PREFACE.

the advantage of students, and those who had the charge
of his manuscripts were naturally inclined to adopt a course
of which they believed he would himself have approved.

The design of reconstructing the Treatise was therefore
abandoned; and it was resolved that the original volume
should be reprinted, and that the manuscripts should be
collected and published separately. This plan has the ob-
vious recommendation of enabling those who are already
familiar with the original work to turn their attention
readily to the new investigations. It will be seen that
many of the Chapters of the present volume may be re-
garded as independent essays or memoirs which lose nothing
by being separated from the other volume ; and indeed no
indications had been left by Professor Boole of the place
which such Chapters were to occupy in the enlarged edition.

I have printed all the unpublished matter relating to
Differential Equations which I found among Professor Boole's
papers. In a few cases it will be seen that an investigation
is incomplete ; such investigations have however been in-
cluded in the volume, because I was unwilling that anything
should be lost which so great a mathematician had written
on a subject he had long and carefully studied.

I trust that no serious error will be found in the volume,
and that any faults which may be detected will be excused
on account of the nature and difficulty of the task that had
to be performed. Many of the manuscripts had not been
finally revised ; some of them were very obscure and had
to be carefully and laboriously copied for the press. In
general the equations were not numbered, and thus only

PEEFACE. Til

blanks occurred in place of references; this circumstance
often caused great trouble and perplexity: I hope however
that a satisfactory result has been finally attained.

I may state for the benefit of those who are conversant
with the first edition of the original work that the theo-
rem which in the present volume is cited as contained in
Chap. II. Art. 1 will be found in Chap. IV. Art. 2 of
the first edition: the change was made by the direction of
Professor Boole's interleaved copy. It was judged conve-
nient to number the Chapters in the present volume in con-
tinuation of those in the original work.

All additions of my own are enclosed within square
brackets. The sheets have been read by the Rev. J. Sephton,
Fellow of St John's College, as well as by myself, and the
volume is much indebted to his care and accuracy. Obvious
mistakes in the manuscripts were of course corrected; thus,
for example, the table at the end of the volume was calcu-
lated by Mr Sephton, because the table in the manuscript
was rendered erroneous by the use of a wrong sign in a
formula.

I. TODHUXTEE.

ST JOHN'S COLLEGI,
Nwtmber, 1865.

LIST OF PROFESSOR BOOLE'S WRITINGS.

In tlie Philosophical Transactions.
On a General Method in Analysis, 1844, pages 225... 282.

On the Comparison of Transcendents, with certain applications
to the Theory of Definite Integrals, 1857, pages 745. ..803.

On the Theory of Probabilities, 1862, pages 225.. .252.

On Simultaneous Differential Equations of the First Order in
which the Number of the Variables exceeds by more than one the
Number of the Equations, 1862, pages 437. ..454.

On the Differential Equations of Dynamics. A sequel to
a Paper on Simultaneous Differential Equations, 1863, pages
485... 501.

On the Differential Equations which determine the form of the
Roots of Algebraic Equations, 1864, pages 733. ..755.

In, llie Transactions of the Royal Irish Academy.

On the Analysis of Discontinuous Functions. Vol. 21, 1848,
pages 124... 139.

On a certain Multiple Definite Integral Same Vol., pages
140. ..149.

In tJte Transactions oftlie Royal Society of Edinburgh.

On the Application of the Theory of Probabilities to the Ques-
tion of the Combination of Testimonies or Judgments. Vol. 21,
1857, pages 597. ..653.

In tJie Bulletin de VAcademie...de St Peterslourg.

Consid6rations sur la recherche des integrates premieres des
6quations differentielles partielles du second ordre, Vol. iv. 1862,
pages 198. ..215. [See page 143 of the present volume.]

In Crelle's Journal fiir Mathematik.

Ueber die partielle Differentialgleichung zweiter Ordnutjg
Rr + Ss + Tt+U (s* - rt) = V. Vol. 61, pages 309. ..333.

LIST OF PROFESSOR BOOLE 3 WRITINGS. IX

In the Cambridge Mathematical Journal.

Researches on the Theory of Analytical Transformations, with
a special application to the Reduction of the General Equation of
the Second Order. VoL 2, 1841, pages 64... 73.

On Certain Theorems in the Calculus of Variations. Same
Vol., pages 97... 102.

On the Integration of Linear Differential Equations with Con-
stant Coefficients. Same VoL, pages 11 4... 11 9.

Analytical Geometry. Same Vol., pages 179. ..188.

Exposition of a General Theory of Linear Transformations.
VoL 3, 1843, pages 1...20, 106. ..119.

On the Transformation of Definite Integrals. Same VoL,
pages 216. ..224.

Remarks on a Theorem of M. Catalan. Same Vol., pages
277. ..283.

On the Transformation of Multiple Integrals. VoL 4, 1845,
pages 20.. .28.

On the Inverse Calculus of Definite Integrals. Same VoL,
pages 82... 87.

Notes on Linear Transformations. Same Vol., pages 167. ..171.

On the Theory of Developments. Same VoL, pages 2 14... 223.

In the Cambridge and Dublin Mathematical Journal.

On the Equation of Laplace's Functions. VoL 1, 1846, pages
10. ..22.

On the Attraction of a Solid of Revolution on an External
Point. VoL 2, 1847, pages 1 ... 7.

On a certain Symbolical Equation. Same VoL, pages 7... 12.

On a General Transformation of any Quantitative Function.
Vol. 3, 1848, pages 112. ..116.

The Calculus of Logic. Same VoL , pages 1 83 ... 1 98.

On a General Theorem of Definite Integration. Vol. 4, 1849,
pages 14... 20.

On the Theory of Linear Transformations. Vol. 6, 1851, pages
87. ..106.

X LIST OP PKOFESSOE BOOLE'S WEITINGS.

On the Reduction of the General Equation of the ntYl Degree.
Same Vol., pages 106... 11 3.

Letter to the Editor of the Journal. Same Vol., pages 284, 285.

Proposed Question in the Theory of Probabilities. Same Vol.,
page 286.

On Reciprocal Methods in the Differential Calculus. Vol. 7,
1852, pages 156. ..166, and Vol. 8, 1853, pages 1...24.

In the London, Edinburgh, and Dublin Philosophical Magazine...
Third /Series.

Remarks on the Rev. B. Bronwin's Method for Differential
Equations. Vol. 30, 1847, pages 6... 8.

ISTote on a Class of Differential Equations. Same Vol., pages
96, 97.

Remarks on a Paper by the Rev. Brice Bronwin, On the
Solution of a particular Differential Equation. Vol. 32, 1848,
pages 413. ..418.

Remarks on a Paper by the Rev. Brice Bronwin, On the Solu-
tion of a Particular Differential Equation. Vol. 33, 1848, page 21 1.
Notes on Quaternions. Same Vol., pages 27 8... 280.

In the Fourth Series of the same Magazine.

On the Theory of Probabilities, and in particular on Mitchell's
Problem of the Distribution of the Fixed Stars, Vol. 1, 1851,
pages 521... 530.

Further Observations on the Theory of Probabilities. VoL 2,
1851, pages 96. ..101.

An Account of the late John Walsh of Cork. In a letter
from Professor Boole to Professor de Morgan. Same Vol., pases
348.. .358.

Solution of a Question in the Theory of Probabilities. Vol. 7,
1854, pages 29... 32.

Reply to some Observations published by Mr Wilbraham in
the Philosophical Magazine, Vol. 7, p. 465, on the Theory of
Chances developed in Professor Boole's ' Laws of Thought.' Vol. 8,
1854, pages 87.. .91.

LIST OF PEOFESSOR BOOLE'S TVEITISGS. XI

On the Conditions by which the Solutions of Questions in the
Theory of Probabilities are limited. Same Vol., pages 91... 98.

Further Observations relating to the Theory of Probabilities
in reply to Mr Wilbraham. Same Yol., pages 175, 176.

On a General Method in the Theory of Probabilities. Same
YoL, pages 431... 444.

On certain Propositions in Algebra connected with the Theory
of Probabilities. Yol. 9, 1855, pages 165... 179.

On a Question in the Theory of Probabilities. By A. Cayley,
Esq. [This paper embodies some observations by Professor Boole.]
Yol. 23, 1862, pages 361... 365.

On a Question in the Theory of Probabilities. Yol. 24,
1862, p. 80.

Separate Publications.

An Address on the Genius and Discoveries of Sir Isaac
Newton. Lincoln, 1835.

The Right Use of Leisure. London, 1847.

The Mathematical Analysis of Logic, being an Essay towards
a Calculus of Deductive Reasoning. Cambridge, 1847.

The Claims of Science. London, 1851,

An Investigation of the Laws of Thought, on which are
founded the Mathematical Theories of Logic and Probabilities.
London, 1854.

The Social Aspect of Intellectual Culture. An Address de-
livered in the Cork Athenseum — Cork, 1855.

A Treatise on Differential Equations. Cambridge, 1859.

A Treatise on the Calculus of Finite Differences. Cambridge,
1860.

[This list contains all Professor Boole's writings which
have fallen under the notice of the editor ; it is possible that
there may be a few omissions.]

CONTENTS.

CHAPTER

XIX. ADDITIONS TO CHAPTER II

XX. ADDITIONS TO CHAPTER VII. ..... 7

XXL ADDITIONS TO CHAPTER VIII ....... 9

XXII. ADDITIONS TO CHAPTER IX ....... 38

XXIII. ADDITIONS TO CHAPTER X. ..... .46

XXIV. ADDITIONS TO CHAPTER XIV. ..... eg

XXV. ON SYSTEMS OF SIMULTANEOUS LINEAR PARTIAL DIFFER-
ENTIAL EQUATIONS OF THE FIRST ORDER, AND ON ASSO-

CIATED SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS 74

XXVI. HOMOGENEOUS SYSTEMS OF LINEAR PARTIAL DIFFERENTIAL

EQUATIONS ......... 90

XXVII. OF NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS Off THE

FIRST ORDER ........ 96

XXVIII. PARTIAL DIFFERENTIAL EQUATIONS OF THE SECOND OBDEB 119

• XXIX. ON THE SOLUTION OF THE PARTIAL DIFFERENTIAL EQUATION
Sr+Ss+ Tt + U(s2-rt) = V, IN WHICH K, S, T, IT, V ARE

GIVEN FUNCTIONS OF x, y, z, p, q ..... 145

XXX. ADDITIONS TO CHAPTER XVII. . . . . . 175

XXXI. THE JACOBIAN THEORY OF THE LAST MULTIPLIER . . 200

XXXII. THE DIFFERENTIAL EQUATIONS OF DYNAMICS [FRAGMENT] 218

XXXIII. ON THE PROJECTION OF A SURFACE ON A PLANE . .221

I

CHAPTER XIX.

ADDITIONS TO CHAPTER II.

1. [!N Chapter n. Art. 9, two methods are given for
solving the differential equation

(ax + by + c} dx + (ax + I'y + c) dy = 0.]

But there exists another transformation by which the equa-
tion may be reduced to, (because it may be constructed from),
an equation in which the variables are separated.

Assume as this equation

(Ay + C] dx'+ (A'x'+ C') dy = 0 ...... (1)

and let x — x + w^y, y = x + m^y.

It will be seen that in these equations united we have as
many constants as in the original equation. Now on substi-
tuting in the assumed equation the values of x and y', and
comparing with the equation given, we deduce a system of
relations equivalent to the following, viz.:

The quantities m^ , m^ are roots of the quadratic
am* — (b + a) m + V = 0.

The quantities A, A', C, C' are determined by the system
of equations

A + A' = a, C+C' = c,

^ 4 A'm^ = a, Cm^ + C'm^ = c'
B.D.E. II.

IS

2 ADDITIONS TO CHAPTER II. [CH. XIX.

from which we find

— a cm, — c

A'=

m9-m1 m^-m,

am, — a cm, — c

— m.

Now (1) gives on dividing by (Ax' + C') (Ay + C} and
integrating

-^ log (A'x' + C'} + \ log (Ay + (7) = const.,

* ' -^*-

or (A'x' + C'} T (Ay1 + C}2 = const.,

which on substitution and reduction gives

i
{(am1 — a'} (x + m1y] + cml — c'^r"'

{(amz — a) (x + mzy) + cmz — c}ami-a'

2. Under certain circumstances the general solutions of
differential equations of the first order fail. This happens in
the above example if m2 = m1, the solution then reducing to

1 = const.

The theory of the deduction of the true limiting form of
the solution in such cases requires a distinct statement.

Let the supposed general solution be represented by
u=C,

C being the arbitrary constant and u a function of x, y, and
constants which are not arbitrary. Suppose too that when
one of these constants k assumes a particular value «, the
function u reduces to a constant v. Then we have

u — v C — v

/C ~~ /C /C *~~ /C

Now the second member being a function of an arbitrary
constant is equivalent to an arbitrary constant and may be

ART. 2.] ADDITIONS TO CHAPTER II. 3

replaced by C. The first member is a vanishing fraction, the
limiting value of which is (-wj, the brackets being used to

denote that after the differentiation k is to be made equal to K.
Hence the solution becomes

In applying this theory to the reduction of the general
solution (2) in the case in which m1 = mt , it must be
observed that the numerator of the first member is the same
function of ml , x, y, as the denominator is of m^ , x, y ; or
attending solely to their functional character with respect to
wi1? w2, we may affirm that the numerator is the same function
of 7nt as the denominator is of m£. Representing these func-
tions by ^(mj, <j>(m2) respectively, we have

But m1, ra2 being roots of a quadratic equation may be
represented in the form

ml = m + k, mt = m — k,
the roots becoming equal when k = 0. Hence

<f> (m + k]

u =

— — ; .

<f> (m — k]

Therefore since

m + k} d<j> (m — k) _ dfy (m — k

dk dm dk dm

we have

- k}
dk - -

1—2

ADDITIONS TO CHAPTER II. [CII. XIX.

7 j / - 5 - — - 5 --

. ,, fdu\ r dm dm

therefore -77

dkj

(7
Thus the solution "becomes on putting C for — ,

40

or --- Jog |7ayn _ a'\ (x + my\ _j_ cm _ c'l = C.

dm am — a

3. [The next Article seems to have been intended to ap-
pear in the enlarged form of Chap, n.; but I cannot discover
what precise position it would have occupied. I conjecture
that " the above demonstration" refers to Chap. II. Arts. 2, 3;
and I have accordingly supplied a reference to equation (3) of
Chap. ii.

I had myself drawn Professor Boole's attention to Chap. II.
Arts. 2, 3. The geometrical process of Chap. n. Art. 3, ap-
pears to have been first given by D'Alembert in his Opus-
cules, Vol. iv. p. 255. D'Alembert calls it a demonstration; it
seems to me only an illustration, at least in the brief form of
the text : and that such was Cauchy's opinion may perhaps
be inferred from the elaborate investigation given by Moigno,
to which Professor Boole refers in Art. 5 of the present
Chapter.

I had also drawn Professor Boole's attention to the state-
ment at the end of Chap. n. Art. 12, that only one arbitrary
constant was involved. Accordingly Article 5 of the present
Chapter developes this statement, and Article 4 sesms intended
to bear on the same subject.]

4. In the above demonstration the relation between y and
x is regarded as one of pure magnitude, and the interpreta-
tion of the differential equation becomes a limiting case of
that of the equation of finite differences (Eq. (3), Chap. n.).
But if we represent x and y by the rectangular co-ordinates

ART. 5.] ADDITIONS TO CHAPTER II. 5

of a moving point on a plane the differential equation may be
interpreted directly. For supposing it reduced to the form

we see that the direction of motion is constantly assigned as
a function of the co-ordinates of position. The entire motion
is therefore determinate as soon as the initial point is fixed.
The result of the motion is a line or curve wholly continuous
or subject to irregularities according to the nature of the func-
tion f(x, y). That the arbitrariness of origin is geometri-
cally equivalent to the appearance of a single arbitrary con-
stant in the relation connecting x and y may be shewn thus.

Let y = $(•*<>, y*,*}

be the relation between x and y indicated by the supposed
motion, x0, ya being the initial point of departure. Then this
point being on the line of motion, x0. y0 are particular values
of a; and y, so that we have from the above equation

which establishes a relation between x0 and y0, and shews
that there exists virtually but one arbitrary constant.

5. It is proved in Art. 3, Chap, n., that the constants
a*0, y0, initial values of the variables x, y in the solution of
the differential equation of the first order, are necessarily
equivalent to one arbitrary constant. I shall shew from the
form of the above solution that this a priori condition is
actually satisfied.

Developing the expression for y [see Eq. (30) of Chap, n.]
in ascending powers of cc, we have

........ (32)

the summation extending from n = r to w = cc. Formin

6 ADDITIONS TO CHAPTER II. [CH. XIX.

lience the differential coefficients of Ar with respect to cr0 and
y0, and reducing by (28), we shall find

. dAr , , . dAr

whence in particular

y\ dA« = A

Eliminate "between these equations f1(x0, ?/0), and we have

Therefore, by Prop. I., Ar is a function of A0, so that tlie
solution reduced to the form (32) contains but the single
arbitrary constant A0.

It remains to notice that the solution must be applied
only under the conditions of convergency, i.e. under the con-
dition that the ratio of the wtb to the (n — l)th term tends to a
limit less than unity as n tends to infinity. For a discus-
sion of the failing cases of this test see ' Finite Differences,'
Chap. v. Generally it is desirable, in order to secure rapid
convergency, to divide the interval x — <r0 into separate equal
portions, to each of which the general theorem of solution
may be applied. If x — x0 be very small the theorem may
be approximately represented by

y -#o=/0»e >300-*o) •

On these principles Cauchy has founded remarkable methods
of solution, which deserve attention from the commentary on
the limits of error on their application by which they are
accompanied (Moigno, Vol. II. pp. 385 — 434).

CHAPTER XX.

ADDITIONS TO CHAPTER VII.

1. [THIS Article relates to Art. 2 of Chap, vn.]

The sense in which (9) may "be said to constitute the
general solution of the differential equation is this. We
obtain from it

giving any particular value to C this will geometrically
represent a curve consisting of two branches, and giving to
C every possible value we obtain an infinite system of such
curves, each consisting of two branches. The aggregate of
branches thus obtained is evidently the same as the aggre-
gate of curves given by the two primitives (5) and (6), un-
restricted by any connexion between cl and c2. In this sense
then the solution (9) is general, that it includes all the parti-
cular relations between y and x which are deducible from
the original primitives (5) and (6). And it is only in this
sense not general that it groups these relations together in a
particular manner.

To the expression of the complete primitive a certain
variety of form may be given without affecting its generality
in the sense above affirmed. Thus, if to the solutions of the
component differential equations we give the forms

ye^-c^O, logy + ax-cz=0,

we should have, by the same procedure, as the expression of
the complete primitive,

(ye™ — c) (log y + ax - c] = 0,

8 ADDITIONS TO CHAPTER VII. [CH. XX.

an equation which may equally with (9) be regarded as the
complete primitive of the differential equation given, and
which in geometry represents the same totality of branches of
curves as (9), with this difference only, that they are differ-
ently paired together.

2. [This Article relates to Art. 3 of Chap, vn.]

The question will here naturally arise, Since if F= c be
a solution of one of the component differential equations,
f(V) = c, in which f (V] denotes any function of F, is also a
solution, by Chap. IV. Art. 3, why not give to the complete
primitive the form

or the stricter form

in which /^FJ, ,^(F2), ...y»(Fn) denote arbitrary functions
of Fj, F2,..., Fn respectively — stricter because the presence of
arbitrary constants and functions in the previous form is a
superfluous generality? It is replied that though the form
just given is analytically more general than (15), it is not
more general than (15) with such freedom as is permitted
in the interpretation of the arbitrary constants. In a physi-
cal or geometrical application we should not only be per-
mitted to assign a particular value to the arbitrary constant
in (15), so deducing what in reference to its source would
then be termed a particular primitive, but to combine the re-
sults of different determinations of c together, so as to obtain
every form of solution which is implied either in the func-
tional equation (F}> or in its component primitives

V —c V = c V —c

r 1 *"! > 'a — c 2 ' ' ' » ' » ~ °n •

The same considerations justify us in speaking of (15) as
the complete primitive, and not as a complete primitive.

CHAPTER XXI.

ADDITIONS TO CHAPTER VIII.

1 . [THE Singular Solutions of Differential Equations of
the First Order received great attention from Professor Boole,
and the Chapter devoted to that subject is one of the most
valuable and important in his work. He continued his re-
searches after the publication of his first edition, and intended
to reconstruct the Chapter with great improvements in the
second edition. After carefully examining the manuscripts I
came to the conclusion that it would be very difficult to re-
write this portion of the work so as to connect the old matter
with the new ; and thus it seemed best to reprint the original
Chapter vin. with corrections of obvious misprints, and to
print the matter intended for the revised form in the present
volume. The plan gives rise to some repetition; but this
seems unimportant, compared with the advantage of preserv-
ing in the author's own language all that he left on an in-
teresting and important point which he had carefully Studied.

2. It may be of service to the student to reproduce the
substance of some remarks on his Chapter vin. which were
sent to Professor Boole soon after the publication of his first
edition ; tor there is evidence in his manuscripts that he paid
great attention to such remarks while engaged in the revision
of his work, and thus the reason and the meaning of some of
his additions and changes may be made more obvious. These
remarks will occupy the next Article,

3. The two pages beginning with " And these conditions
are sufficient/' and ending with "do not lead to conflicting

10 ADDITIONS TO CHAPTER VIII. [CH. XXT.

results" forming part of Arts. 3 and 4 of Chapter vin., seem
obscure and difficult. The following may perhaps be substi-
tuted with advantage.

The only ways in which

dy ^ df(x, c] and dy = df(x, c} df(xt c) dc
dx dx dx dx dc dx

can be equivalent when c is variable, are

fi\ -u df(x-> c) A
(1) when -' ;=Q,

/^N i . c)

(2) when -'^ ;=co;

in the latter case -~ = GO , and therefore -=- = 0. and this
ax dy

implies that the singular solution is of the form x = constant.
Thus there can be no singular solutions except such as

df (x c]

are found from — ---,-- — - = 0, and such as are found from
dc

x = constant.

Similarly, if the complete primitive be expressed in the
form x = P(y, c), there can be no singular solutions except

such as are found from - -j2 — = 05 an<i sucn as are found

dc

from y = constant.

In Art. 8 of Chapter vin. we read, " We may pass over
the case in which the above equation is satisfied independ-
ently of c, because the relation obtained would involve x

{JT)

only, while it is a condition accompanying the use of -f- = oo

that it leads to solutions involving y at least." It is ob-
jected, Why may we pass over this case? Such a case might
occur and furnish a solution, and then we should want to
know the character of that solution. Take for example

p = xny ; here if n is negative, -j- is infinite when x = 0, and

y ^n + I

this is a singular solution. For the general solution isy=cen+l,

ART. 3.] ADDITIONS TO CHAPTER VIII. 11

and so x=Q is not a case of it. The words — ichile it is
a condition... at least — seem very difficult, for by supposition

we are now investigating what is furnished by -~ = oo .

Professor Boole met the objection in substance thus :

" It will be found that the rules in the book are correct in
this case. "What is implied in the Chapter, though not stated

with sufficient clearness, is that if -f- = oo leads to a solution

dy

which does not involve y in its expression, nothing is to be
inferred whether it is singular or not. Then the proper test is

" 1'

\pj

i - =co.
ax \

In this example we have

-Z. = co gives x* = GO ; no inference ;
dy

dx\~>= y

Hence x =- 0, provided n is between 0 and — 1 , or y = 0.

Consider these separately :

First. Let n be between 0 and— 1, and x = 0. This is
by the test a singular solution. Substituting it in the com-
plete primitive we get y = c, which confirms this.

Second. Let y = 0. This satisfies the differential equa-
tion; but from the fact that it comes from -7- (-] = oo we

ax \p)

have no inference ; from the fact that it does not come from
-±- = GO we have the inference that it is a particular integral : it
corresponds to c = 0.

12 ADDITIONS TO CHAPTER VIII. [oil. XXI.

There remains the case of x=0 when n is between — 1

and — co . As this does not satisfy -=- ( - ) = co , we infer that
* J dx \p)

it is a particular integral. To prove this we have

When x = 0 this gives, since 1 +. n is negative,

c = co or c = — co ,

according as y is positive or negative. This is like Ex. 2 of
Chap. vin. Art. 8."

The remark made by Professor Boole in the above reply,
that if — = GO leads to a solution which does not involve j

nothing is to be inferred... is important. It corrects the state-
ment put too strongly in Chap. vill. Art. 7, " All we can affirm

is that if ~- = co gives a solution at all it will be a singular
solution."

From Art. 8 onwards it seems assumed that a solution for
which — = 0 is always to count as a singular solution, even if

it should coincide with a particular integral. This does not
seem to have been quite the view of the former part of Chap-
ter vni. : see Arts. 5 and 6 of the Chapter.

In Ex. 3 of Art. 9 we read, " the second is obviously a
singular solution." This means that since we have a solu-

tion which makes -f- infinite, we conclude that it is a singular
ay

solution.

So in Ex. 5 of Art. 11 we read, " is evidently a singular
solution," when it seems better to say, " and is therefore a
singular solution."

4. The additional matter relating to Chapter vm. begins
with another example which was to be placed at the close of
Art. 3 of that Chapter.]

ART. 4.] ADDITIONS TO CHAPTER VIII. 13

Ex. The differential equation

has for its complete primitive

\x* + y* — m* — y — c = 0.

T-T d(h y dd> x

Here y- = . y — - 1, -f- = (

/HI »/^ I „.* »v,» /7-T? i/^." I ^.

<7y V JJ +y*—m* dx *Jx* + y* — m*

Hence -- = -- - = - - -

/ , ,

- V J72 -f -

771

Both -^ and — vanish then if
dc dc

st»+ /-*?=*<>.

This therefore is the singular solution and it satisfies "both
the tests, as both x and y are contained in its expression.

Of the partial tests

d4> d<b dd>

__ — O !— — m — -r>

^ v> J — J J — *' l

dc dx ay

the first is not satisfied, the last two are satisfied.

The determination of c as a function of x by the solution

df(x c\
of the equation J ^ — = 0 is equivalent to determining

what particular primitive has contact \vith the envelope at
that point of the latter which corresponds to a given value
of x.

One important remark yet remains. The elimination of c
between a primitive y=f(x, c) and the derived equation
dy

— = 0, does not necessarily lead to a singular solution in the
< c

14 ADDITIONS TO CHAPTER VIII. [CH. XXL

sense above explained. For it is possible that the derived
equation

dc

may neither on the one hand enable us to determine c as a
function of x, so leading to a singular solution ; nor, on the
other hand, as an absolute constant, so leading to a particular
primitive. Thus the particular primitive

dtj
being given, the condition -~- gives

eex = 0,

whence c is + co if a; be negative, and -co if a? be positive.
It is a dependent constant. The resulting solution # = 0
does not then represent an envelope of the curves of particu-
lar primitives, nor strictly one of those curves. It represents
a curve formed of branches from two of them. It is most
fitly characterized as a particular primitive marked by a sin-
gularity in the mode of its derivation from the complete pri-
mitive.

All the foregoing observations and conclusions may be
extended to the case of solutions derived from the condition
dx _
dc

5. We have seen that the equation -~ — 0 may be satisfied

by an absolutely constant value of c, so leading to a particu-
lar primitive and not a singular solution. In this case
-v/r (x + h, c) as well as i|r (x, c) would vanish, and the nume-
rator of (9), instead of being the difference of a finite and an
infinite quantity, would be the difference of two infinite and
equal quantities. [Sec Chap. vill. Art. 8.] It would not there-

fore be infinite. Hence we conclude that -*- would not become

ay

infinite for a particular primitive in the strict sense of that

ART. 5.] ADDITIONS TO CHAPTER VIII. 15

term, i. e. for a solution derived from the complete primitive
by giving to c an absolutely constant value.

This is one point of contrast between the conditions
*y_0 ^=ao

-, - VT, 7 - vAJ •

dc ay

There is another not less important. As the numerator
of (9) may become infinite not only when -fy (x, c) = 0, but
also when ty (x, c} = infinite, we see that a relation between

y and x which makes ~ infinite will not necessarily satisfy

the differential equation. On the other hand, it is not a par-
ticular primitive in the strict sense of that term.

dx
Exactly in the same way the condition -^ = 0, as relating

to the complete primitive, leads to the condition

d (\_
-

as relating to the differential equation, with the same points of
difference in the respective applications.

dti m~l

Ex. Let -2- = myn , and suppose m a positive constant

greater than 1.

dp -

Here ^=(m -!)/»,

which becomes infinite when y = 0. As this involves y and
satisfies the differential equation it is a singular solution.

To confirm this conclusion we may refer to the complete
primitive

y=(x-cr,

which does not give y = 0 for any particular value of c.

Now let m be a positive constant less than 1. We have
still — = co when y = 0 ; but this value of y no longer satis-

16 ADDITIONS TO CHAPTER VIII. [CH. XXI.

fies the differential equation. It is not a solution at all, nor
would it result from the application of the condition ^ ;- = 0

to the complete primitive. The distinction of character of
the two tests is here made manifest.

6. We may express the most important results of the
foregoing investigations in the following theorem.

THEOREM. Every solution of a differential equation of the
first order which is derived from the complete primitive by
giving to c a variable value will, if it involve y in its expres-
sion, satisfy the condition

*-«;

dy

and if it involve x, the relation
d /I

-r- 1-1=00.

ax

But relations satisfying these conditions will not neces-
sarily be solutions of the differential equation.

In applying this theorem the following points must be
carefully attended to.

1st. No conclusion can be drawn from the satisfying of

the condition ~ = GO when the relation in question does not

dy
contain y in its expression, nor from the satisfying of

d /IN

7T ~ =co
ax \pj

when the relation in question does not involve x in its ex-
pression. For these conditions being respectively derived

from -if- = 0 and - ~ = 0 are subject to the same limitations
dc dc

in their application.

2ndlv. It may be that -f- or -=- ( - ) assumes for a particu-
dy dx \pj

lar relation between x and y the indefinite form n . In this

*

ABT. 6.] ADDITIONS TO CHAFiEll VIII. 17

case we must seek by the development of its terms or by
other known modes its true limiting value or values. Finite
values will indicate particular primitives, infinite values sin-
gular solutions, and when such values emerge together out of
the same relation between the variables, the solution will be
a particular primitive possessing the geometrical properties of
a singular solution. Its locus will be a particular curve en-
veloping other curves of the same family.

See Examples 2 and 3 of Chap. vin. Art. 11.
We have seen that the conditions

iln d f\\

77- = QO» j- [- =:0
ay dx \j)J

indicate in general the existence of a relation between c and
x or c and y. And when that relation is such as to enable us
to determine c as a continuous function of one of the vari-
ables, the corresponding solution of the differential equation
is singular, and is geometrically represented by an envelope
of the curves of primitives. But it may be, as we have seen
in a particular example, that the relation does not determine c
as a function of x or y ; but according to the language already
used, c is a dependent constant, or in some other way different
from the constant of an ordinary particular primitive. Let
us examine in particular instances the kind of singularity
which may hence arise.

Ex.1. Given

Here -f- = - (1 + log y).

dy x ^ ° dl

This becomes infinite if x = 0 ; but this not involving y
must be rejected. Again, it becomes infinite if y = 0, and
this proves to be a solution of the differential equation, the
limiting value of the indeterminate function in the second
member being 0 (Todhunter's Differential Calculus, Chap. x.).
Xow the complete primitive is y — e", discussed in Art. 4.
The constant c is there shewn to be dependent, the solu-

B. D. E. II. 2

18 ADDITIONS TO CHAPTER VIII. [cil. XXI.

tion y = 0 emerging from the complete primitive by making
c = — co if x be positive, and c = co if a; be negative.

Ex. 2. Given f-^J — xy ~- + y2logy=Q.

xii + ?/ (x'2— 4 loe: y)b
Here p = -^^ — — - j

3fl

^P _ •'K ± (»2 — 4 log g)_ j. 1

therefore =

. , ,,

(a-8 -4 logy)4

and this is made infinite by y = 0 and by a;2 — 4 log T/ = 0,
i.e. by

Both satisfy the differential equation.
Now the complete primitive is

y = <?*-*.

We see at once therefore that the second of the above solu-
tions is singular. The first however is deducible from the
complete primitive by making c = co or c = — co , irrespec-
tively of the sign or value of x, provided only that x be
finite ; not so however if x be infinite. The value of c is not
therefore in the most absolute sense independent of that of x.
If from the complete primitive we seek the singular solution

by the condition :J = 0, we get the two equations

The second of these determines c as a function of x, and
leads to the second of the solutions obtained above. The first,
though it does not determine c as a function of a*, still ex-
presses a relation between c and x, which is the ground of
the fulfilment of the condition

dp

/ =00.
dy

ART. 6.] ADDITIONS TO CHAPTER VIII. 19

We may further notice a peculiarity arising from this rela-
tion. Supposing x finite and the solution y = 0 a particular
integral, it presents the singularity that it is the only case in
which two particular integrals agree. We might in any com-
plete primitive, by changing c into c2, get two values of c for
the same particular integral, but then it would be for every
particular integral.

One negative character seems indeed to mark all the cases
in which a solution involving y in its expression satisfies the

condition -^ = <x . It is that such solutions do not emerge
dy

from the complete primitive by the attributing of a single and
absolutely constant value to c. The relation which makes •—
infinite satisfies the differential equation only because it satis-
fies the condition -f- = 0, and this implies a connexion be-
ac

tween c and x, which is the ground of a real though it may
be unimportant singularity in the solution itself.

At this point, then, the question arises, whether the term
singular solution shall be confined to that class of solutions,
the loci of which represent the envelopes of curves of primi-
tives, or shall be extended to all solutions which, satisfying the

condition -~ = co , indicate the existence of a relation be-

dy

tween c and x, and possess an actual singularity arising from
this source. While the all but universal consent of mathe-
maticians is in favour of the former course, it is to be remem-
bered that the question is solely one of definition. Xot such
is the question how singular solutions of the envelope species,
or as would more generally be said true singular solutions,
are to be distinguished from all other solutions. This we
now propose to consider. The question is not an isolated one.
It stands in close relation to a series of properties of singular
solutions which admit of an orderly development.

2—2

20 ADDITIONS TO CHAPTER VIII. [CII. XXI.

Discrimination of singular solutions of the envelope species.

7. A negative test, which in the great majority of cases
suffices for the present object, is suggested by the following
consideration.

dit
The differential equation determining -£- as a function of

d*y dsi/

x and y determines also ~~ , -rjj , ad inf., and the know-
ledge of these enables us to construct in a developed form
the complete primitive. See Chap. n. Art. 12.

fit I Ci tl

The values of -jr- , -r4 » &c. ad inf., as derived from the

CtJfr C13C

differential equation, are the same as those derived from the
complete primitive.

But a solution deduced from the condition -j- = <x> is only

ay

7 9

constructed so as to yield the same value of -y- as the given

differential' equation does. If it be of the envelope species,
the curve it represents has in general no continuous contact
with the curve of any particular primitive. It will not there-
fore generally yield the same values for , ••{ , -7-^ , &c. as

the differential equation does. It will not therefore generally
satisfy the differential equations of an order higher than the
first, which would be derived from the given equation by dif-
ferentiation. Hence we have the following Proposition.

PROPOSITION. If a relation which makes -J- infinite satisfy

ay

the given differential equation of the first order, but do not
satisfy all the higher differential equations obtained from it,
such solution will be singular and of the envelope species.

ART. 7.] ADDITIONS TO CHAPTER VIII. 21

Ex. 1. By comparison with its complete primitive we saw
in Art. 5 that -^- = my *" has for a singular solution y = 0
when m is a constant greater than 1.

We will first suppose m a fractional quantity greater than
1, and endeavour to deduce the character of the solution with-
out making use of the complete primitive.

From the solution we have

g-O, g-0,*c. «?,»/.

But from the differential equation

g-(— i)^|-«{«-i)^,

and generally

<Fy m~T

~jr = m(m-l)...(m-r + l)y'*.

Hence, when r is less than m, the substitution of y = 0 gives

as before. But if r is greater than m, it gives

d'y

^ = CO'

We conclude that the solution is of the envelope species.
Secondly, suppose m a positive integer greater than 1.

In this case we find, when r is less than m, the same series
of values as before • but for r = m we have

and this also shews the solution to be of the envelope species.

22 ADDITIONS TO CHAPTER VIII. [CH. XXI.

Ex. 2. The differential equation

!+/
is satisfied by

Is this a singular solution or a particular integral ?
From the solution we find

dy x d2y _ 4
dx ~ y' dx2 y* '

From the differential equation we shall have

dx2 2 (y - xp)

ClI/

substituting in which the value of -jr- , obtained from the
proposed solution, we find

da? 2y* y3 '

Now this differing from the value before obtained, we con-
clude that the solution is singular and of the envelope species.

And this result is verified by comparing the solution with
the complete primitive

As the test above exempl fied is merely negative, it is in-
sufficient. For it is conceivable that an enveloping curve
should have an infinite order of contact with each of the curves
which it envelopes, and this is also possible. Any test found-
ed upon a comparison of the values of differential coefficients,
any test therefore furnished by the Differential Calculus, would
be insufficient for the discrimination of such cases.

Ex. 3. Given -j- = y (log y)\

ART. 8.] ADDITIONS TO CHAPTER VIII. 23

Here -^- = 00 gives y = 0, and this satisfies the differen-
tial equation.

From this solution we find

From the differential equation we have

which consists of y multiplied by a rational and entire func-
tion of logy. It is easy to see that all the higher differential
coefficients of y hence derived will possess the same character.
And all such vanish with y.

We can therefore neither affirm nor deny that the proposed
solution is of the envelope species.

8. Before demonstrating a general Rule for the discrimi-
nation of solutions of this character, we shall notice certain of
their properties which serve to indicate in what direction the
Rule is to be sought. [See Chap. vili. Art. 14.]

As the exact differential equation differs from the sup-
posed given differential equation by having acquired a factor
which the singular solution makes infinite, so the given dif-
ferential equation may be said to differ from the correspond-
ing exact one by containing a factor which the singular solu-
tion makes to vanish. If we knew that factor, we could by
rejecting it reduce the given differential equation to a form in
which it would no longer be satisfied by the singular solution.
Now Poisson has shewn on a particular assumption, which
does not however affect the principle of the demonstration,
that this factor can be found when the singular solution is
known. His demonstration is in substance as follows.

Let us represent the given singular solution of the dif-
ferential equation by

w = 0,
u being a given function of x and y. Then introducing u and

2i ADDITIONS TO CHAPTER Till. [fH. XXI.

x instead of y and x as variables, the differential equation
after transformation will assume the form

du ... .

5 -/(<*>«).

Now this equation being satisfied by w = 0 and the first
member vanishing, the second must also. Poisson now
assumes, and the assumption must be carefully noted, this
second member to be capable of being developed in ascending
positive powers of u. Supposing it so developed, the diffe-
rential equation becomes

in which A, B,... are functions of x, and a, /3,... ascending
positive indices.

Hence if u = 0 be a singular solution we have, putting p
„ du

f°r^'

-/- = Jaw-1 + Spue-* + &c. = « .
an,

But this demands that there should be at least one nega-
tive power of u in the development in the second member.
Therefore a — 1, the lowest index, must be negative. There-
fore a being already positive must lie between 0 and 1.

"We may give therefore to the transformed differential
equation the form

du _

a being a positive fraction, and Q not vanishing with w.
Hence, dividing by M°,

du

ART. 9.] ADDITIONS TO CHAPTER VIII. 25

a differential equation which is not satisfied by u = 0, since
u = 0 gives w1"" = 0, and the first member vanishes while the
second member does not vanish. In its present form then
the equation is not satisfied by u = 0. We see also that the
property of being satisfied by u = Q has been lost not in
reality through a transformation, but through the rejection of
an algebraic factor ua from the transformed equation. It has
been shewn in the treatment of Clairaut's equation, how in
the ascent by differentiation to an equation of a higher order
a somewhat analogous effect is produced, the singular solu-
tion emerging out of a factor of that higher equation.

If we inquire what is essential in Poisson's demonstration,
we shall find it to consist in that the transformed equation is
of the form

in which while Q neither vanishes nor becomes infinite when
u = 0, the functions

both vanish with u. The question whether U is of the form
ua as Poisson supposes, or is not, is wholly immaterial .
This will fully appear from the demonstration of the follow-
ing theorem, which is in effect Poisson's freed from arbitrary
assumptions.

9. PROPOSITION. If u = Q be a solution of a differential
equation of the first order between y and x, and

du . .
fa =/(*,«)

represent the form which that equation assumes when u and x
are assumed as variables instead of y and x, then if f(x, u) be
resolved into two factors Q, U, of which Q neither vanishes
nor becomes infinite when u = 0, while the functions U and

-jj both vanish when u = 0, then the differential equation can
be reduced to a form in which it shall cease to be satisfied by

26 ADDITIONS TO CHAPTER VIII. [CH. XXI.

In the statement of this proposition x is supposed to be
constant in the integration relative to u.

The differential equation after the transformation which
introduces u and x as variables becomes

£-«*

Let

so that v is in general a function of x and u, the form of
which is known by integration when that of U is given.
And again, transform the differential equation by making v
and x the variables instead of u and x. We have

fdv\ _ dv dv du
\dxj dx du, dx'

in which -=- is the differential coefficient of v with respect to

(1 'V

x, on the above hypothesis as to the constitution of v as a

function of x and u. while IT ) is the differential coefficient

\dxl

on the hypothesis that v is reduced to a function of x alone
by the conversion of u into a function of x.

dv 1 du

omce ~T = ^TTI ~T~

(i?/ i J ci'jr

the above equation becomes

Now if u = 0 give v = 0 for all values of x, it will there-
fore give

-*

and further,

^w_ _J p ^f_A
dx~dx)0 (J~Q>

ART. 9.] ADDITIONS TO CHAPTER Till. 27

since we are permitted to make u = 0 before effecting the
differentiation with respect to x. Hence the equation re-
duces to

0=0.

And this is not satisfied, since by hypothesis Q does not
vanish with u.

Hence if u = 0 make I -~ = 0, the transformed differen-
tial equation will no longer have u = 0 for a solution.

COR. Assuming Q = 1, which does not violate the hypo-
thesis respecting Q, and gives

U=f(x, u),

we see that if

— = f(x it]

be satisfied by u = 0, and if at the same time u = 0 gives

du _0

(x M) ~~

the differential equation can be transformed so as to cease to
admit of the solution u = 0.

It is obvious however that it is best to assume Q so as to
make the subsequent integration for determining v the sim-
plest possible.

It is manifest that a solution which can thus be made to
cease to satisfy the differential equation cannot be a particular
primitive. For the complete primitive of the transformed
differential equation which it does not satisfy is convertible
into the complete primitive of the original differential equa-
tion which it does satisfy, merely by writing therein for v its
expression as a function of x and y. It cannot therefore be a
case of the complete primitive in any sense. It must be
a singular solution of the envelope species.

The converse proposition still remains to be proved.

28 ADDITIONS TO CHAPTER VIII. [CH. XXI.

10. PROPOSITION. If u = 0 be a singular solution of thf>
envelope species of a differential equation of the first order, and
if by assuming u and x as the variables, the differential equa-
tion is reduced to the form

du j.. .

T*=^x' «)•

then will

du

,/(»,*)

become 0 when u = 0.

Let the complete primitive be represented by

F(x, u) = C,
then, since

dF(x, u) dF(x, u) du _
dx du dx

we have if for brevity we represent F (x, u) by F,

du
dx

therefore *

f(x,u)

dx

Now w = 0 being a singular solution, F(x, 0) is not a con-
stant ; for if it were, the complete primitive would, on giving
to C the constant value in question, yield u = 0 as a particu-
lar primitive. And this would equally be the case whether
that constant were finite or infinite in value. We see then

that F(x, 0) must be a function of x, and therefore --A— ' — -

must either be a function of x, or a finite constant differing
from 0 ; the latter if F(x, 0) be of the form ax + b, the former

ART. 10.] AUDITIONS TO CHAPTER VIII. 29

dF(x, u)
if it be not of that form. Therefore the value of -

dx

when M = 0, since in this we are permitted to make u = 0
before differentiating with respect to x, will be a function of
JL-, or a finite constant differing from 0.

Now it is manifest that in general

' 1 dF j TrTdF ,

— — — a u = H\ -j- du,
at du J0 du

dx

0

where H is some value intermediate between the greatest and

least values which —TV assumes within the limits of intesra-
ar

dx

tion. When these limits are, as in the above case, infinitesi-
mal, we have

jr ^ o .

Hence

dx
du

But we have seen that — —^ — - does not vanish. Hence

its reciprocal, the first factor of the right-hand member of the
above equation, does not become infinite. Again,

F(x, u)-F(x, 0)
vanishing when u = 0, we have

du

when u is made infinitesimal as was to be shewn.

30 ADDITIONS TO CHAPTER VIII. [CH. XXI.

It will "be observed that the previous general expression
for I -j- . becomes infinite if u = 0 is a particular integral.

* 0 J \ ' **/

For then, F(x, 0) being a constant, - ' vanishes, while

GLJC

F(x, u) —F(x, 0) does not vanish so long as u differs by
however small a quantity from 0.

These propositions form the ground of the following Rule
for the discrimination of singular solutions of the envelope
species from all others.

11. RULE. The proposed solution being represented by
u = 0, let the differential equation, transformed by making u
and x the variables, be
du

Determine as a function of x and u the integral

du

in which U is either equal to f(x, u), or to f(x, u) deprived
of any factor which neither vanishes nor becomes infinite
when u = 0. If that integral tend to 0 with u the solution is
singular.

Ex. 1. Determine whether y = 0 is a singular solution or
particular integral of the differential equation

Here, since u — y, no preliminary transformation is needed.

r du i

We have ., -y ... = — —

•/0y(loS*/) iogy

which tends to 0 with y. Hence the solution is singular.
To verify this we observe that the complete primitive is

ART. 11.] ADDITIONS TO CHAPTER VIII. 31

and this cannot be reduced to y = Q by giving any constant
value to c.

We have seen in Art. 7 that the test -which is founded upon
the comparison of differential coefficients does not suffice to
characterize the above solution.

Ex. 2. The equation ~ - = — — is satisfied by y = 0.

CLJC> J.

Is this solution singular or particular?

Here also no transformation is required. We have, reject-

ing the factor - which neither vanishes nor becomes infinite
x

when =

[

— — = log log y — log log 0
, los: v

and this being infinite, however small y may be. may properly
be said to tend to infinity as y tends to 0. The solution is
therefore particular.

It will perhaps appear at first sight as if in the above ex-
ample we ought to write

when y is made equal to 0. But the course of the demonstra-
tion shews that the value of the definite integral must be first
obtained on the hypothesis that u (in this case replaced by y]
is finite, and then the limiting value which its expression
approaches to, as u approaches to 0, be sought. And in this
•case, since for all finite values of u however small the integral
is infinite, its limiting value is infinite.

The complete primitive in the above case is

y = f,

and the nature of the solution y = 0 has already been dis-
cussed in Art. 4.

32 ADDITIONS TO CHAPTER VIII. [cil. XXI.

History of the Theory of Singular Solutions,

12. It is remarkable that while the theory of enveloping
curves and surfaces was at once founded and developed by
Leibnitz in 1692 — 4*, the corresponding theory of the singular
solutions of differential equations has been of very slow growth.
The existence of these solutions was first recognised in 1715
by Brook Taylor; it was scarcely more than recognised by
Clairaut in 1734. Euler, in a special memoir, entitled Expo-
sition de quelques Paradoxes dans le Calcul Integral, published
in the Memoirs of the Academy of Berlin for 1756, n'rst made
them a direct object of investigation ; but the foundations of
their true theory were only laid in 1768 in his Institutiones
Calculi Integralis. Laplace, Lagrange, Legendre, Poisson,
Cauchy, and De Morgan have in various ways developed and
extended that theory; but there has been so remarkable a
want of unity and connexion in this long series of researches,
that important portions of the theory appearing in a too
isolated form have been neglected, forgotten, and rediscovered.
I purpose here to give a brief account of what seems most cha-
racteristic, rather than of what is most original in their several
researches ; for the germs of nearly all subsequent discoveries
on the subject are to be found in the great work of Euler.

Taylor and Clairaut appear to have been led by accident to
the noticing of singular solutions ; the former while directly
occupied on the solution of differential equations, the latter
while discussing a remarkable class of problems relating to
the connecting properties of different branches of the same
curve. Taylor gave them the name singular, while Clairaut,
and Euler too in his memoir, regarded them as a species of
paradox, not merely from their non-inclusion in the general
integral, but from the mode of their discovery through a
process of differentiation. The memoir of Euler, though it
sheds no light on the real nature of these solutions, contains

• Ada Eruditorum, 1692, p. 168 ; 1694, p. 311. Opera, Tom. in. pp. 2G4,
296.

Methodus Jncrementorum, p. 26.

Mtmoirea de CAcadcmie des Sciences, 1734, p. 209.

ART. 12.] ADDITIONS TO CHAPTER VIII. 33

an interesting theorem concerning their connexion with the
form of the differential equation, viz. If this equation can
be brought to the form

Vdz=Z(Pdx+Qdy],

in which z is a function of x and y, and Z of z, then will

Z=0

be a singular solution. In his Institutiones Calculi XktegraKs,
Tom. i. p. 393, however, Euler gives a rule which is the
counterpart of that of Cauchy. [See Chap. vin. Art. 12.]
He shews that if u = 0 be a particular integral, and if the
differential equation be reduced to the form
du , .

, du

then

I

* o

d> (x, u]

The limits of integration are here supplied. The reasoning,
which is not fully developed, is the following. From the
transformed equation we have

du

„ i du
Hence x =

<j> (x, u) '

x _ If du

~7r ~ * + "TV

If this be satisfied by a solution involving x and y, and if
that solution be a particular integral, then on putting for x
its value in terms of u and integrating, the above equation
will be satisfied by giving some particular constant value to
C. But if the supposed particular integral be u = 0, then x
and M being independent, we may perform the integration
with respect to u as if x were constant. The resulting equa-
tion cannot be free from x unless C be infinite, and then it

B. D. E. n. 3

34 ADDITIONS TO CHAPTER VIII. [CH. XXI.

evidently not "be satisfied unless I — c be infinite.

J <l>(x, u)

We infer then that this is a necessary condition in order that
u = 0 may be a particular integral.

This is Euler's fundamental theorem, and from this, by
means of an hypothesis agreeing with that of Poisson con-
cerning the form of the transformed differential equation, he
arrives at the condition

dp

— = &.
dy

[In the passage to which Professor Boole refers, Euler
does not undertake to discuss the nature of any solution,
but only of a solution of the form x = constant. On his
page 408 Euler proceeds to discuss the nature of any solu-
tion. Professor Boole seems to me to attribute too much
to Euler. For the convenience of those who wish to ex-
amine the original, I will give the reference to the passages
in the later editions of Euler's Institution's Calculi Integralis :
Vol. I. pages 343 and 355 of the edition of 1792 ; Vol. I.
pages 342 and 354 of the edition of 1824.]

Laplace in the Memoirs of the French Academy for 1772,
p. 343, established the tests

dp d fl\

-f- = co , 3- - = co ,

dy dx \pj

and shewed their respective uses. He established also the
test which consists in the comparison of differential coefficients,
and he supposes it universal. He adopts the hypothesis of
his predecessors as to the forms of expansion, but with some
recognition of its insufficiency.

Lagrange in the Memoirs of the Academy of Berlin for
1774, p. 197, and 1779, p. 121, appears first to have developed
the theory of singular solutions in its two forms of derivation
from the complete primitive and derivation from the differen-
tial equation, and to have established the essential connexion
of these. But supposing the differential equation to be ex-
pressible in the rational form

ART. 12.] ADDITIONS TO CHAPTER VIII. 35

and employing the differential coefficients of F(x,y,p) in-
stead of those of p he was led to sacrifice rigour to symme-
try. One of his results has often since been adopted as a
test of singular solutions. It may be thus stated.

PROP. A singular solution makes the general value of

cPu

-=5 , deduced from the differential equation in its rational and

air

integral expression, to assume the form - .

[The demonstration is given in Chap. vill. Art. 14.]

This ambiguity of value of -^ is evidently but an expres-
sion of the fact that the contact of a curve of the complete
primitive and that of the singular solution is not in general
of the second order.

The result given in equation (5) of Chap. vill. Art. 14 has
also been adopted as the test of singular solutions.

The researches of Poisson and Cauchy have already been
noticed. It is certainly remarkable that the final test to
•which Cauchy's analysis led should be essentially the same as
that which had been discovered by Euler so long before.

Professor De Morgan has thrown an important light upon
the nature of the conditions

dp dp

j =00» ;/ =GC>
dy ax

which are fulfilled by all singular solutions in the expression
of which x and y are both involved. He has shewn that any
relation between x and y which satisfies these conditions will

73

satisfy the differential equation unless it make -~ , as derived

from the differential equation, infinite ; that it may satisfy the

ji

differential equation even if it make -7 \ infinite ; lastly, that

CUE

3—2

36 ADDITIONS TO CHAPTER VIII. [CH. XXI.

if it do not satisfy the differential equation, the curve it
represents is a locus of points of infinite curvature, usually
cusps, in the curves of complete primitives.

The proof is as follows :
Let p = $ (x, y]

be the differential equation. Then the proposed conditions
are

d$ (x, y] d<f> (x, y)

—fry- ' ~^~ '

therefore by differentiation,

__ ---

dxdy dy2 dx dx* dxdy dx

whence we have

dy _ dx dy _ dxj
dx

dy* dx dy

These are two equivalent expressions for the same value of
-,— . The question now is, under what circumstances this

CtiC

value of ~ will satisfy the differential equation.

Cm?

Now from that equation we have by differentiation

d*y _ d<$> d<f> dy
dx* dx dy dx'

whence

d*y d<f>
ily _ dx* dx
dx d$

dy

ART. 12.] ADDITIONS TO CHAPTER VIII. 37

If th
infinite,

If then -T-? be finite we have, since fr and -j- are both
dx1 dx dy

dy _ dx

dy

and this by the rule for the evaluation of fractions of the

form ^ is equivalent to the value in either of its forms before

obtained for -/-. Hence, any relation which satisfies the
ax

given conditions and makes -~ finite, will satisfy the diffe-
rential equation.

And the same result holds even if —^ be infinite, provided

that ~~ -r- -3T- vanish.
aar dy

Lastly, as when this result does not hold, the failure is due

72

to the infinite value of -^ , we see that the line in which the

locus of the proposed relation intersects the curves of primi-
tives will be a locus of their points of infinite curvature.

[Transactions of the Cambridge Philosophical Society,
Vol. ix. Part H.]

Legendre's ^Memoir of 1790 throws but little light upon
the subject of this Chapter. But it exhibits the theory of
the singular solutions of differential equations of the higher
orders, both ordinary and partial, in a form of great beauty,
and will be noticed in the proper places.

( 38 ) .[CH. xxii.

CHAPTER XXII.

ADDITIONS TO CHAPTER IX.

1. BY successive application of the second theorem of
Chap. IX. Art. 13, a linear equation of the ntb order may be
reduced to one of the (n — r)th order, if r distinct integrals of
what the given equation deprived of its second term would
be are known.

The reduction may however be effected immediately by
the method of the variation of parameters. In this and in
most general investigations connected with differential equa-
tions great advantages in point of brevity and of the power of
expression are gained by the employment of the symbol of
summation S, and of the language of determinants. I shall
exemplify this here.

Suppose the given equation to be

and let yl} ya,'»yr be r particular values of y, satisfying the
equation

Thus y = c^ + c^ . . . + cryr

is a solution of the latter equation including these particular
solutions. We shall represent this by

(3),
and regarding the quantities clt c2,...cr, represented here by

ART. 1.] ADDITIONS TO CHAPTER IX. 39

cf as variable parameters, shall seek to determine them so that
the above value of y may satisfy the equation given.

These r parameters, enabling us to satisfy t — 1 arbitrary
conditions, besides satisfying the differential equation, we may
choose these so that

dy cPy cTly
dx' da?*'

may be the same inform as if ct, ca, . . . cr were constant. Xow
from (3)

whence

provided that the condition

be satisfied. Differentiating the first of these equations, we
find in the same way that

provided that the condition

2 d?h dci _ 0
dx dx

be satisfied. And thus continuing we see that the system of
r equations

will hold true provided that the r — 1 conditions

40 ADDITIONS TO CHAPTER IX. [CH. XXII.

be satisfied. In each of these equations the symbol £ is to
be interpreted by giving to i the successive values 1, 2,... r,
and taking the sum of the results.

Differentiating the last of the equations (4), we have

ty. = s & 4- * ^y* — *

dxr idxr ' * dx'-1 dx'

As we cannot impose the condition that the last term of
this equation shall vanish, let z represent its unknown value,
then

%-*%+> ..................... ««•

Now the system of equations (5), together with

constitute a system of r simple algebraic equations deter-
mining by solution the r quantities

dc^ dc2 dcr
dx' dx'"' dx

in terms of their coefficients and of z, and therefore in terms
of x and z, since the coefficients are known as functions of
x. It is evident also that as the second members of all the
equations but one vanish, and the second member of that is
z, the values so determined will be of the form

X^X^... Xr being known functions of x. Thus the r un-
known quantities /y1v -y^ are made to depend upon only

one unknown quantity, viz. z. It remains then to deter-
mine z.

For this purpose we must complete the expression of the
differential coefficients of y, and substitute in the given dif-
ferential equation, and then seek to satisfy that equation.

ART. 1.] ADDITIONS TO CHAPTER IX. 41

Xow differentiating (6) we have

i V <fyi <ki , dz

1 • djJ dx "*~ dx

<c
on substituting for -7-' the value XjZ as above determined.

(••V

We observe that the coefficient of z is here a known function
of a\ If we differentiate this equation and in the result sub-

dc

stitute as above for -, , we shall have a result of the form
dx

L and J/ being known functions of x. Ultimately then
we have

* * n K~T

Q

dx* l dx* dx ' ' dx" '

Thus, while y and the differential coefficients of y up to the
(r — I)01 are of the same form as if c,, cs,... cr were constant,
the succeeding ones differ in containing an additional portion
consisting of z, and differential coefficients of z multiplied by
known functions of x. The result of substitution of these
values in the given differential equation will therefore consist
also of two classes of terms, viz: terms under the sign of
summation, which will be the same in form as if cx, c2,...cr
were constant, and terms involving the differential coefficients
of z up to the (n — r)th, with multipliers which are known
functions of x. We shall in fact have

^-r^ +

42 ADDITIONS TO CHAPTER IX. [CH. XXII.

Now yi being by hypothesis an integral of (2), the first
line of the above equation vanishes, and there remains the
linear equation of the (n — r)th order

Supposing z hence determined, we have in general

r

and hence

y = yAXlzdx + yAX.izdx +yr\XTzdx,

and as z will have n — r distinct values, each involving an ar-
bitrary constant, the above equation will furnish n — r distinct
values of y, each involving an arbitrary constant. It is to be
observed that no arbitrary constant need be added in the inte-
gration of the terms X& dx, for the effect of such addition
would only be to reproduce the known integrals c^. In
this way, however, the equation would represent the general
integral of the differential equation given.

2. Let us examine the form of the result in the particular
case in which r=n — l.

Here we have

dxm' ldxn
from m = 0tom = n— 2, then

ART. 2.] ADDITIONS TO CHAPTER IX. 43

Accordingly the differential equation for z will be

Now the equations for determining

dx

<

become on putting Xp for 4p , and writing for brevity y\ for

Whence, by the theory of determinants,

^ Mfyf*''

J/ standing for the determinant

«

2/3

(*-«)

44 ADDITIONS TO CHAPTER IX. [CII. XXII

is ultimately a
dr*y{\ I dM

Now the determinant is ultimately a function of x ; and
such indeed that

'daTlJ M dx '
For

i, ^y

'

_ ___

dx ** dyt dx dyl dx ' dy£n^ dx .

Now M being homogeneous and of the first degree with
respect to the quantities y1, y2, ..... yjt_2, we have

Hence 2 -=— y- is what M becomes when in its expression

V\->y*i •••yn_i are changed into y/, 7/2', ... y'n_v therefore it is
what M becomes when two of its rows of elements become
identical ; therefore it vanishes. In like manner all the other
sums in (8) vanish excepting the last, for y^n~l\.... y,^n~l] is
not a row of elements of the determinant M. Thus we have

^ dM (n_1}_

Hence

, 1 dM

Thus the equation (7) becomes
dz /I dM

therefore z = -i €~fA^ f Me^** Xdx.

M J

ART. 2.] ADDITIONS TO CHAPTER IX. 45

Hence, since

dci_ -,?• _ 1
= iZ =

whence

v f1 dM j
we have y = % J ^ ^^ 2^r,

2 being given above.

In the case of X = 0, vre have

whence

( 46 ) [CH. xxiii.

CHAPTER XXIII.

ADDITIONS TO CHAPTER X.

1. THE theory of singular solutions of differential equations
of the higher orders has been presented in the most complete
form which it has yet received by Legendre. (Memoires de
V Academie Eoyale des Sciences, 1790, p. 218.) He determines
first the possible forms of these solutions considered as emerg-
ing from the complete primitive by the variations of its arbi-
trary constants, and secondly the theory of their derivation
from the differential equation itself. I shall follow the same
order, and shall in the end endeavour to point out in what
respect Legendre's theory may be regarded as complete, and
in what respect it is imperfect.

Suppose the differential equation to be of the nih order,
and let it when solved with respect to the highest differential
coefficient of y be represented by

yn=$(x>y>yny*T~y^ .................. (i),

in which, for brevity,

dy d?y dny

ni — ". ni — _ !? ni — - tL

2/1 dx> *• dot'"'** XT'

Let also its complete primitive, solved with respect to y,
be represented by

y=f(x, alt oa,...ow) ..................... (2),

ap o2, ... an being the arbitrary constants of the solution. If
we differentiate (2) with respect to x, regarding ax, a2, ... an
no longer as constants but as functions of x, so to be deter-
mined as to leave the expressions for y^ y2, ... yn as functions

ART. 1.]

ADDITIONS TO CHAPTER X.

47

of al, a.,. ... «„ the same as before, we shall have, on repre-
senting the second member of (2) by/,

•whence

df df da^ df da, df dan

dx dav dx da, dx " dan dx '

provided that

df dal df <Jan df dan _

Differentiating on the same hypothesis the first of these two
equations, we find in the same way

I U' J

y* = ^?'

provided that

d?f dal d^f da, d?f dan

dx da. dx dx da, dx" dx da dx

1 z n

And continuing thus, it results that the system

._£ ,=#..,»#.. ..f«

dx ' dx^ ' dx*

will be satisfied, i.e., that yl, y,, . . . yn will have the same ex-
pressions when at, a2, ... an are variable as they have when
these are constant, provided that the law of their variation be
determined by the conditions

da. dx da. dx " da. dx

JL Z

* J- *^ 2 i • * n ___ A

da^ dx dx da, dx dx' ' dan dx dx

da_i d"f da, d*f dan _
dx da. dx*~l dx ' ' da. dx"'1 dx J

48 ADDITIONS TO CHAPTER X. [CH. XXIII.

In this system the coefficients of

oa1 daz dan

dx ' dx ' dx

are known functions of x, a^ , a2 , . . . an when the form of f is
known.

Eliminating

dal da2 dan
dx ' dx ' ' dx '

we have a relation between x, ax, a2, . . . an; and this relation,
with the given complete primitive and the first n — 1 of the
derived and reduced equations, viz., with

_/ -df

will enable us to eliminate a1? «2, ...«„, and to obtain a rela-
tion of the form

dy d2y dn

This is a differential equation of the (n — l)th order. It dif-
fers in its origin from the given differential equation, in that
a new relation between x, al5 a2, . . . an has been employed in
place of the nih equation, derived by differentiation from the
complete primitive, for the elimination of the constants.

The differential equation of the (n — l)th order thus obtained
has an integral expressing y in terms of a1, and n — 1 arbi-
trary constants. This is the most general form of a singular
solution of the differential equation.

It is possible that the elimination of al , a.2 , . . . an may
lead to a resulting differential equation which, instead of
being of the order n — 1, is of the order n — 2, n — 3, &c.
The complete integral of such equation would be a singular
solution of the differential equation. These possible types of
solutions are distinguished by Legendre according to the
number of arbitrary constants which they contain. A solu-

ART. 1.] ADDITIONS TO CHAPTER X. 49

tion containing n — 1 arbitrary constants is called by him a
singular solution of the first order ; one containing n — 2 ar-
bitrary constants a singular solution of the second order, and
so on.

Adopting this language we might term the complete primi-
tive a singular solution of the order 0.

Lastly, any relation between x and y, which satisfies the
given differential equation, will constitute a particular case,
either of the complete primitive or of one of the general
forms of singular solutions above defined. In the case of
differential equations of the first order it is seen that no arbi-
trary constant can appear in the expression of the singular
solution.

Ex. The equation

V^-a.

3) (dx X

has for its complete primitive

ax*
y=— + bx+a* + b* .................... (6),

required its singular solution.

Proceeding as above, we find on the hypothesis of a and b
being variable parameters, the same formal expressions for

-~ , -r^ as if those parameters were constant, viz.

dx

•(7),

provided that the variation of a and b be such as to satisfy
the conditions

da db

dx dx

B. D. E. II.

50 ADDITIONS TO CHAPTER X. [CH. XXIII.

Eliminating hence -j- and — , we have

a;
2a - — - 2bx = 0.

2

And from this, the complete primitive, and the first of the
derived equations (7) eliminating a and b, we find

This is the differential equation of the first order, by the
solution of which the most general form of the singular solu-
tions of the given differential equation will be determined.

Reducing it to the form

x .

and integrating, we find

This then is the general expression for the singular solu-
tions of the given differential equation. We see that it in-
volves in its expression one arbitrary constant.

The differential equation (9) may properly be termed a sin-
gular first integral of the given differential equation. The
singular first integral (9) has itself also a singular solution,
viz.

1 . 1 4

rt/ — __ _ sy*£ __ _ /W* •

y- 4* 1&x

but this is not a solution of the original differential equation.
Nor have we any right to expect that it should be so. A
singular solution of a differential equation of the first order
does not necessarily satisfy the differential equations of higher
orders derived from that equation, Chapter xxil. Art. 7.

2. It remains to establish the theory of the derivation of
the singular solution from the differential equation without
the mediation of the complete primitive.

ART. 2.] ADDITIONS TO CHAPTER X. 51

Resuming the differential equation in its reduced form (1),
and representing its second member by <f>, suppose an infini-
tesimal variation given to the arbitrary constants of its com-
plete primitive, and let the symbol 8 be used to denote the
corresponding derived variations of y, yl} . . .ya. Then we
have

and so on. Hence, substituting and transposing,

d'Sy d* d-l8y d* <Z"-% . _ . .

~dx* ~dy^ "dx^ ~d^ ~d^ '

Let us consider the real nature of this equation.

If a value of y, suppose y = ^r(x), satisfy the given differ-
ential equation, that value substituted in the coefficients

of the above equation •will convert them into functions of x,
and the equation itself will become a linear differential equa-
tion, the solution of which will determine $y as a function of
x. If the differential equation (10) be really, as it is appa-
rently, of the ?ith degree, §y will have n arbitrary constants,
O, . . . a, and will be of the form

j, P8, . . . Pn being functions of x. Hence

p
I

We see thus "that the given solution y=TJr(x) will be a
articular case of this general integral involving n constants.
t will therefore be a particular integral of the proposed.

4—2

52 ADDITIONS TO CHAPTEE X. [CH. XXIII.

If, owing to the constitution of its coefficients, the differential
equation (10) be of the degree n — 1, we shall have

- n-i )

and y = ^r(x] will then be a particular case of a solution
involving n — 1 arbitrary constants. It will therefore be a
singular solution of the first order. Even so, if the differen-
tial equation (10) be of the degree n — 2, y = ty(x) will be a
singular solution of the second order. And generally, if
the differential equation be of the r& degree, y = ^(x) will be
a singular solution of the order n — r.

Kesuming the equation (10) it is evident that it cannot

be of the degree n — 1, unless -r-^- be infinite. For, dividing

dy^

by 7 , we have

__
d<f> ' dxn dxn-* C'

in which the first term does not vanish unless ~— be infi-
nite. This then is the necessary condition for a singular
solution of the first order. For one of the second order we
must have in like mariner

d<j> d(j)

and so on.

It follows hence that to find the singular solutions of a
differential equation of the nih order, we ought to differentiate

the equation, regarding y, -^ , -^ , &c. as varying through

the variation of the arbitrary constants, to form in this way
a linear differential equation for Sy, to examine the conditions
tinder which this equation reduces to the (n — l)th, or to a lower
degree, and to examine whether the most general relation be-

ART. 3.] ADDITIONS TO CHAPTER X. 53

tween x and y which satisfies such condition, satisfies also the
given differential equation. If so it may be regarded as a
singular solution.

Resuming the last Example, viz.
dy 1 ^d*y_ (

and operating with 8 we have

which reduces to a linear differential equation of the first
order for determining By, provided that we have

dy

=0.

Eliminating -y^ from the given equation by means of this
there results

dy

and we find on differentiating this that it does constitute a
solution of the given equation. It is therefore a singular
first integral of that equation. We see that it agrees with
the result obtained under the same name in the previous
Article, and the rest of the solution need not be repeated.

3. Upon Legendre's theory, and upon its results, the fol-
lowing observations may be made.

1st. We learn from it that there may exist different

eneral forms of the solution of a differential equation of the

n01 order, viz. the complete primitive involving n arbitrary

54 ADDITIONS TO CHAPTER X. [CH. XXIII.

constants, and general forms of singular solutions containing
fewer than n arbitrary constants. A solution y = ty (x) of
unknown origin being given, we construct a differential equa-
tion for determining By, and, solving it, form the expression
for y + By, and from the number of infinitesimal arbitrary
constants it contains, determine the nature of that general
value of y of which the given value is a particular case.
Now we are not to infer from this that the form of y + By will
be the same as the general value of y in question. But we
may infer that it will be a form to which that general value
is reducible. And the actual reduction will be effected by
expressing the general solution (as is always possible) in a
form permitting its expansion in ascending powers of the
arbitrary constants, and in the expansion making these con-
stants infinitesimal, and rejecting all powers of them above
the first. In fact, if

y=f(x, a,, a

2,

be any general form of solution which, when we assign to
al} «2,...ar particular values (e.g. make them vanish) re-
duces to

then we shall have

y + By = +(x) + Ba, + &%... + Bar,

the brackets denoting that after differentiation we make
at , a2 , . . . ar vanish.

This is that limiting form of the solution which Legendre's
method enables us to construct by the solution of a linear dif-
ferential equation ; and the ground of the sufficiency of his
method consists in this, that the infinitesimal quantities

Ba1} 8aa, ... Bar)

which are in fact the arbitrary constants of that solution, are
equal in number to the arbitrary constants of the general
unlimited solution, the nature of which is thus made known.

ART. 3.] ADDITIONS TO CHAPTER X. 55

2ndly. Legendre's tests for differential equations of the
higher orders are in kind and effect analogous to the tests

dp d 1

-T-=00, -=- - = CO

ay ax p

for differential equations of the first order. They enable us
to decide whether a solution possesses singularity, not whether
it possesses the envelope species of singularity. The comple-
tion of Legendre's theory would consist in the discovery of
those further tests dependent upon integration which corre-
spond to the test of Euler and Cauchy for differential equa-
tions of the first order.

[CH. XXIV.

CHAPTER XXIV.

ADDITIONS TO CHAPTER XIV.

[Art. 1 was intended to follow Chap. xiv. Art 2.]

1. As the condition of dependence of functions of two
variables is of fundamental importance in connexion with the
theory of ordinary differential equations, so the generalized
condition of dependence of functions of any number of vari-
ables forms a fundamental part of the theory of partial differ-
ential equations. This is contained in the following proposi-
tion.

PROP. I. If

... un are functions of #,, a?2, ...#n,

but are as such so related that some one of them is expressi-
ble as a function of the others, or more generally that there
exists among them some identical equation of the form

F(u1,u,,...un)=0, ................... (1),

BO that as functions of a^, #2, ...#n they are not mutually
independent, then, adopting the notation of determinants, the
condition

Ci T flT (IT

dxv ' dxz ' " " cfo,,

C?Mn </Mn <?Mn

/-j.-v* ' /7-T* /T'T*

UUJ- UU/0 €*•**/-

= 0

•(2),

is identically satisfied. Conversely, if the above condition be
identically satisfied, the functions ut , uz , . . . un are not mutu-
ally independent in the sense above explained.

ART. 1.] ADDITIONS TO CHAPTER XIV. 57

First let it be noticed that the Proposition is but a general-
ization of that of Chap. n. Supposing U and u to be two
functions of x and y, the condition of their dependence is
affirmed to be

dU dU

dx ' dy

du du

dx' dy

i. e. it is the result of eliminating dx, dy, from the equations

dU

rlv. -L.

dx

= 0,

du

du
dy

and therefore it is

dU du dU du _
dx dy dy dx

as expressed in Chap. n.

We proceed to the general demonstration.

Let the first member of (1), considered as a function of
MI? M2, ... un be represented for brevity by F; then differen-
tiating, we have

dF , dF .. dF ,

-j- du. + -7— du + ... + -j— dun = 0,
du^ dua dun

from which it follows that if du^ dua, ... dun_1 are equal to 0,
then is dun equal to 0; or, since MI} w2, ... un are functions of
a?,, «2, ... acn, that if

du

du

du

du

dx,

+

\-^dx = 0
dx* "

^<-i j
, 1 dx. = 0

(3),

58 ADDITIONS TO CHAPTER XIV. [CH. XXIV.

then is

Thus the last n equations, linear with respect to
dxlt dx2, ...dxn)

are not independent, and therefore by the theory of linear
equations the determinant of the system vanishes identically.
Now this is expressed by the condition (2).

It remains to prove the converse, viz. that if the condition
(2) be identically satisfied, the functions ult u2, ...un will not
be mutually independent.

First, the n — 1 functions w , ua, ... u^ are either mutually
independent or not mutually independent.

If not, then the n functions ul , w2 , . . . un are not mutually
independent, and the Proposition to be proved is granted.

If ult w2, ... un_i are mutually independent as functions of
a?!, fl?2, ... xn, they may be made to take the place of n — 1 of
these quantities, e.g. xt, o?a, ... xn_1 in the expressing of un,
i.e. we may, by means of the expressions for w1? w2, ... w^,
eliminate from that of un the quantities xl , x2 , . . . xn_1 , and so
express un as a function of wz, w2, ... un_v and xn. Suppose
this done, then the system (3), (4) will be converted into

dut = 0, du3 = 0, ...... dun-i — ^>

dun , dun , du , du ,

Now, the determinant (2) vanishing, the equations of the
linear system (3), (4) are not independent ; therefore those of
the transformed system, as written above, are not independ-
ent; therefore the last equation of that system must be a
consequence of the others which manifestly are independent.
But from the form of that last equation we see that such can-
not be the case unless we have

ART. 2.]

ADDITIONS TO CHAPTER XIV.

59

which implies that nn is a function of MI? uy, ...*Vi merely.
Hence the functions ut. uz, ... ua are not independent, as was
to be shewn.

The first member of the equation of condition (2) is com-
monly called the functional determinant of MI} ua, ... un with
respect to a^, x3, ... ccn. The proposition may therefore be
expressed as follows.

The condition of dependence or independence of any sys-
tem of functions of as many variables is the vanishing or
non-vanishing of the functional determinant of the system.

On account of the great importance of this proposition it is
desirable to illustrate it by an example.

Ex. Are the functions

x + 2y + z, x — 2y-r3z, 2xy — xz+ tyz - 2z*
mutually independent or not ?

The equation of condition is

1, 2, 1

1, -2, 3 =0,

that is,

- 4 (— x + 4y — 4z) + 8 (2y - z) — 2 (2x + 4z) = 0,

which is identically satisfied. Hence the functions are de-
pendent. In fact, representing them by u} v, w, we have

[Art. 2 was intended to follow Chap. xiv. Art. 4.]
2. As it has been shewn that a primitive

(1)

leads to a linear partial differential equation of the form

(2),

60 ADDITIONS TO CHAPTER XIV. [CH. XXIV.

provided that u = a, v = b, are integrals of the system of ordi-
nary differential equations

dx _ dy _ dz . .

7~~Q~ ti-

lt is evident that we shall obtain a solution of the partial dif-
ferential equation (2) by constructing the system of ordinary
differential equations (3), deducing their general integrals

u = a, v = 5,
and then constructing from these the primitive (1).

But the question arises, Will this be the most general solu-
tion of the partial differential equation given ?

That it will be so, may be shewn by means of the general
proposition. See Art. 1.

For let w = 0 represent any solution whatever of the given
partial differential equation. Differentiating this with respect
to x and y, we have

dw dw dw dw

substituting the values of p and q formed from this in the
given equation, we have

which must be identically satisfied.

In like manner, u = a, v = b being solutions of the same
equation, we find

du du -du

dx dy dz
which must be identically satisfied.

AP.T. 3.] ADDITIONS TO CHAPTER XIY. 61

Eliminating P, Q, E from these three equations, it results
that the functional determinant of 'w, w, v, with respect to x, y, z,
•will identically vanish. Hence w is a function of M and v,
and the equation w = 0 is a particular case of

F(u, v) = 0,

•which is thus shewn to be the general integral of the given
equation.

"We are thus led to the following general Eule.

RULE. To integrate the equation Pp+Qq=R we must
form the system of ordinary differential equations

dx dy _dz
~P = ~Q~R'

deduce their general integrals in the form

u = a, v = b,
and construct the equation

• F(u, v} = 0.

This will be the general solution sought.
[Art. 3 was intended to follow Chap. xiv. Art. 5.]

3. The above theory may be extended to linear partial dif-
ferential equations of the first order, without regard to the
number of the variables.

First, the theory of the genesis of such equations is ex-
pressed in the following proposition.

PROP. A primitive equation of the form

^(Ml,u,,...«=0 (1),

in which wt , vs , . . . un are any given functions of the vari-
ables z, dependent, and a:,, a;,, . . . xn independent, will satisfy
the linear partial differential equation obtained by eliminating
dz, dx:J dxa) ... dxn from

62 ADDITIONS TO CHAPTER XIV. [CH. XXTV.

expressed as total differential equations with respect to the
primitive variables, and the equation

dz -pl dxl — p2 dxa . . . — pn dxn = 0.

Of this important proposition I propose to give two distinct

proofs.

1st proof. Forming the total differential of the given
equation we have, on representing its first member by F,

dF 7 dF 7 dF 7

-j— du. + -j- du, ... + 3— dun = 0.
dul duz dun

Now this cannot be true for all forms of the function unless
we have the separate conditions

Strictly to prove this, suppose F^ F^ ...Fn to be any n
distinct and independent functions of ul,u2,...un, and as
such, distinct and independent forms of F. Then the above
equation gives

df\du df\du df\d
dF0 , . dFn , dF0 ,

,d ^

Now Flt F?,...Fn being independent, their functional
determinant with respect to W1,w2,...wn, does not vanish.
This again is the condition necessary and sufficient that the
above system of linear equations may be independent ; and
this lastly being the case, their only possible solution will be

as was to be shewn.

ART. 3.] ADDITIONS TO CHAPTER XIV. 63

These equations in their developed expression

du. , du. , du. ^ du. 7

•T-I<£EI + — -dxy...+ j-Ldxn + -j-1 dz = Q,

CLU ClU -m f.

— cijc •••••••*••••••••*••*••••••• "T" ~i> — dz == \) y

dx. 1 dz

+ -^dz = 0,

enable us to determine the ratios of dxt, dx3, . . . dxn, dz in
the form

dx1 _ dxt _ dxn _ dz ,9«

x^-x," -^-x"

where X15 -T2,...jrB,^ are functions of the original vari-
ables. And now, forming the equation

and eliminating the differentials, we find

X^ + Xzp2 ...+ Xnpn = R
for the partial differential equation sought.

2nd proof. Differentiating the given primitive with respect
to xv as contained explicitly in the functions ut , u2, . . .un, and
also implicitly in the same through z, we have, on represent-
ing the first member of the equation by F,

dF (du. du\ dF (du. du\

- I - 1 J. /n _ * I _1_ - I - ? -|_ in - - I

du, \dxl +Pldz)+ du, (dx, +Pi dz)

dF /du du

+Pl= '

or

since

dF du, dF du, dF dun dF

L _L ^ I . ™ I _ ay — I)

du dx du dx ' ' du dx dz ^

dF du. dF du. dF dun dF

_ _ * l _. _ . i _ _ '* ^— _

dul dz du% dz " dun dz dz

64 ADDITIONS TO CHAPTER XIV. [CH. XXIV.

Differentiating thus with respect to the remaining inde-
pendent variables, we obtain finally the system

dF du^ dF_
du dx du

dFdu^ dF

* * * 1^ 7 7 i^ 7 r'l ~~ "j

aun ax, dz-^1

dF duv dF duq dF dun dF
du, dx0 dua dx0 ' dun dx0 dz J

dF^dui dF du^ dF dun dF
dul dxn duz dxn ' ' dun dxn dz **

from which, in combination with the equation
dF du. dF dus dF dun dF

,_, ._. * l_ « I ™ — A

du^ dz duz dz ' dun dz dz
we can eliminate

dF dF dF dF

du^ ' duz ' " " dun ' dz '

The result will be

du^ ch^ dun . =0,

ft If* /r/Y9 finf* ~^"

«-*»</*» VV«A/_, \A/VUy.

dUi d\ dun ^

dz ' dz'" ' dz '

or, converting rows into columns,

€tU+ CjuU^ dU+ CLlt*

I 1 I 1

dxl> dxz'" " dxn' dz

flu* dun dun dun

dxt' dxa'' " dxn ' dz

Pn Pv Pnt -1

ART. 3.] ADDITIONS TO CHAPTER XIY. 65

which is the determinant form of the result affirmed in the
proposition.

The second of the above forms of demonstration seems to
be preferable to the first,, in that it rests only upon the consi-
deration of the one general form of the function F. I have,
however, given the two proofs, chiefly in order to illustrate
an important remark, viz. that, in nearly all general re-
searches connected with partial differential equations of the
first order, two modes of procedure, the one involving the
use of differentials, the other that of differential coefficients,
may be employed, and that between the forms to -which these
respective modes give rise, a certain law of reciprocity will be
found to exist.

The theory of the solution of the partial differential equa-
tion

follows immediately from that of its genesis. If we repre-
sent by

«! = «!> ttt=ail--M.= °.»

the integrals of the system of ordinary differential equations
(2) a solution of the given partial differential equation will
be represented by (1). That this will be also the most gene-
ral solution may be shewn by the argument of Art. 1. For
if w = 0 represent any solution, then since

dw dw dw dw

3^t

we find

Ydw

-a. j Sj- ...n-j -T-

1dxl Saor2 dxn dz

from which, in combination with the corresponding equations,

du d^ du^ ,Jfdul_

-A.J — r -A--J— ••• + -i* -j -- r -ft -j -- U,
1 dx * a«c dx dz

Y * j- Y * i.Y»A. 7?» - n

.A. j h -A -j— . . . + A, -= h 21 -j- = 0,

^dx aaa? dx dz

B. D. E. II.

66 ADDITIONS TO CHAPTER XIV. [CH. XXIV.

eliminating Xlt Xz, ... Xn, R we obtain a result which ex-
presses that the functional determinant of w, w1? ... un with
respect to the original variables is virtually 0. Whence w is
a function of w1? %2, ... un, and the proposed solution is in-
cluded in the one to which the above method of solution
leads.

That method may therefore be stated in the following Rule.
RULE. To integrate the linear partial differential equation

X—+X — + X dz =R

1 dx^ 2 dx2 ' ' n dxn

form the system of ordinary differential equations
dxl _ dxz _ dxn _ dz

and deduce their general integrals

then

F(Ul,u2,...un)=0

will be the general integral sought.

[The general observations were intended to follow Chap.
xiv. Art. 6.]

General observations.

4. The relation which exists between a proposed linear
partial differential equation and its auxiliary system of ordi-
nary differential equations should be carefully studied. While
it is proper to say as above that the general integral of the
one requires the knowledge of all the integrals of the other,
it is also proper to describe that general integral simply as
the most general form under which an integral of the auxi-
liary system can appear. If

ui = an W8 = os» ... wn = an
are integrals of that system, then

ART. 5.] ADDITIONS TO CHAPTER XIY. 67

is the one general form of an integral of that system, and
due regard being Lad to the arbitrariness of F, this is equi-
valent to

5. The form which the auxiliary system assumes when
the given partial differential equation is deficient in any of its
terms should be noticed.

If Xl = 0, the auxiliary equation

becomes, on clearing of fractious,

dxl = 0.
And thus, if Xt, Xt,... Xr vanish, the given equation being

X — + X ^Z + Y — = Y
the auxiliary system will be

cLxr+. ctxr,a dxn dz
md the integrals of this system being of the form

ic general solution of the given equation will be

This conclusion would follow also from the principle laid
down in Chap. xiv. Art. 2.

Linear partial differential equations in which the absolute
term is wanting, and which are therefore of the form

xr dz . .^ dz_ Y dz _

5—2

68 ADDITIONS TO CHAPTER XIV. [CH. XXIV.

may be termed homogeneous. As in this case one of the
auxiliary equations is

dz = 0,

the general integral will be

ul} u^, ...... un_l "being found by the integration of the remain-

ing auxiliary equations

dx dx dx

When X^ JT2, ...... Xn do not contain z, the solution is best

exhibited in the form

6. Every linear partial differential equation can be converted
into a homogeneous one containing one additional variable.
For it is shewn in Art. 3, that if u = 0 be any integral of

x.7r + x*-^~- + x«r = x>

1 dx^ z dx2 dxn

then is.

v du , v du . v du , vdu

a homogeneous equation with a new variable.

From the general integral of this equation, that of the
former one may be deduced by making u = 0. .

7. The solution of partial differential equations is some-
times facilitated by introducing a new system of independent
variables. The actual transformation is greatly facilitated
by the following symbolical theorem.

THEOREM. If the partial differential equation

~~
dx

X — 4- X — 4- X ~~ — X

1 z ' ' "

ART. 7.] ADDITIONS TO CHAPTER XIV. 69

be expressed symbolically in the form

Az = X,
in which

A_ Y ^ 4- T — \-Y d

A = JL -j -- r -A-3 T~^ • •• T -A* T — ,

1dxl dx2 dxn

then, if ylt y«, ...... yn be a new system of independent vari-

ables given in expression as functions of the old ones, the
transformed equation will be

For, regarding z as a function of ylt yt, ...... ynt we have

dz_^dz_dy, + dz_dy* dz dyH

dx^ dy^ dxt dyt dxl ' ' dyn dxl '

dz dz dif dz dy, dz dv_

— •"! _j «y 2 i '?* .

, dxn dyl dxn dyt dxn" dyn dxn'

whence, substituting in the given equation we find, as the
total coefficient of ~ , the expression

.

~H ••• T •*** ~i — »
dxa * dxn '

or symbolically, Ayt ; and so on for the other coefficients.
The result then is

dz .. . dz ,,

It remains only after calculation of Ayu Ay2, ...... AyB, as

functions of cc1? xa, ...... xn, to express these functions and JL

intermsofy^y,,. ..... yn.

[It appears from the manuscript that an example was to
have been supplied here.]

70 ADDITIONS TO CHAPTER XIV. [dl. XXIV.

[The next Article may "be considered supplementary to
Chap. XIV. Art. 10.]

Singular Solutions of partial Differential Equations.

8. Legendre's theory developed in Chap, xxill. for ordi-
nary, may be applied also without essential change to partial,
differential equations. Regarding the independent variable
z as receiving an infinitesimal change Bz through infinitesimal
change, not in the values of the independent variables

X-

but in the values of the arbitrary constants of the complete
or in the forms of the arbitrary functions of the general inte-
gral, and performing upon the given equation the operation
denoted by B, we shall obtain a linear partial differential
equation for determining the general value of Bz corresponding
to any particular given value of z. If that linear equation be
of a lower order than the differential equation given, then the
equation expressing the value of z + Bz will be a limiting
form of a solution less complete or less general than the com-
plete or general solution of the differential equation given,
and the given solution, formed by making the infinitesimal
constants in the limiting form actually 0, will be singular.

Conversely, to deduce singular solutions without the know-
ledge of the complete or the general integral, we ought to
construct the equations of condition for the reduction of the
equation determining Bz to a lower order than the equation
given, and the most general solution of the differential equa-
tions of condition so formed, will be the most general expres-
sion for the singular solutions of the differential equation
given.

Ex. (px -qyY<l + ±mx* (z — xp) = 0,

in which

dz dz

AKT. 8.] ADDITIONS TO CHAPTEE XIY. 71

Representing the first member of the equation by Fy we
have, on operating by 8,

dF dSz dF d$z

-j- ^r + -j- -j- + -7- 6z = o,
dp ax dq ay dz

and the conditions

dF dF

~ = 0, °-f- = 0,
dp dq

necessary to reduce the equation for 8z to a lower order give
(px -qy)q- 2mx3 = 0,

From these we find
p =

definite and simultaneous values of p and q, which being sub-
stituted in the given equation lead to

z =

and this, as it gives the same values of p and q as those
obtained before, will necessarily satisfy the given equation.
It is therefore a solution, and from the nature of the analysis,
a singular one.

Legendre shews that this singular solution is also dedu-
cible from the general integral of the given partial differen-
tial equation. That integral is the result of the elimination
of a from the two equations

(a) + az — mxy = 0,
{</> (a) - ax} <f> (a) - 2x$ (a) + z = 0.

To deduce the singular solution he supposes <j> (a) to be not
simply a function of a, but a function of a and of one or
both of the independent variables. He expresses the varia-

72 ADDITIONS TO CHAPTER XIV. [CH. XXIV.

tion of <£ (a) derived from this new source by S, and operating
on the first equation with 8, finds

(a) - 2ax} S<£ (a) = 0 ;
therefore <j> (a) = ax'

Substituting this in the equations of the general integral, and
eliminating a, we find

z — 2m*x%y*
as before.

Legendre states his theory of the derivation of the singular
solutions of partial differential equations from the equations
themselves with great brevity, but still as a general theory.
Arid there is nothing in the statement that carries with it any
apparent restriction upon either the order or the degree of the
equations given. Until however we are in possession of a
perfect theory of the genesis of partial differential equations
we shall not be entitled to say that Legendre's theory of
their singular solutions is a perfect one; for until then we
cannot even define, in a perfectly general way, the nature of
the operation denoted by 6.

[The next three Chapters all relate to the subject of partial
differential equations of the first order. The manuscripts do
not appear to have received their final revision from Professor
Boole. It is certain that he intended the contents of Chapter
XXV. to form a part of the new edition ; and it is highly
probable, although not certain, that the contents of Chapter
xxvi. and Chapter xxvu. were also to be included.

The three Chapters are mainly derived from two memoirs
by Professor Boole, published in the Philosophical Trans-
actions.

The first memoir is entitled On Simultaneous Differential
Equations of the First Order in which tlie Number of the
Variables exceeds by more than one the lumber of the Equa-
tions : it occupies pages 437... 454 of the Philosophical Trans-
actions for 1862.

The second memoir is entitled On the Differential Equa-
tions of Dynamics. A sequel to a Paper on Simultaneous
Differential Equations: it occupies pages 485... 501 of the
Philosophical Transactions for 1863.

The first memoir was finished before Professor Boole had
seen Jacobi's researches, which are cited at the beginning
of Chapter xxvi ; these researches indeed could only just
have been published. In his second memoir Professor Boole
describes Jacobi's methods, refers to his own already pub-
lished, and points out the nature of the connexion between
them.]

( 74 ) [CH. xxv.

CHAPTER XXV.

ON SYSTEMS OF SIMULTANEOUS LINEAR PARTIAL DIFFEREN-
TIAL EQUATIONS OF THE FIRST ORDER, AND ON ASSO-
CIATED SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS.

1. THE term simultaneous is here applied to a system of
partial differential equations, to signify that in that system
there is but one dependent variable, the general expression
of which, as a function of the independent variables satisfy-
ing all the equations at once, is the object of search. All
linear partial differential equations of the first order being re-
ducible to the homogeneous form, we shall presuppose this
reduction here. Under this form indeed the problem actually
presents itself in Geometry, in the theory of partial differential
equations of the second order, and in Theoretical Dynamics.

We are sometimes led, in connexion with the same class
of inquiries, to systems of ordinary differential equations
marked by the peculiarity that the number of the variables
exceeds by more than one the number of the equations. Such
systems are intimately connected with the former — stand
to them indeed in a similar relation to that which the
Lagrangean auxiliary system bears to the single partial dif-
ferential equation from which it arises. The theory which
explains this connexion, and grounds upon it the method of
solution of both systems will form the subject of the present
Chapter.

Connexion of the Systems.

2. PROP. I. The solution of a system of simultaneous
linear partial differential equations of the first order may be

ART. 2.] LINEAR PARTIAL DIFFERENTIAL EQUATIONS. 75

made to depend upon that of a system of ordinary differential
equations of the tirst order in which the number of the vari-
ables exceeds by more than one the number of the equations.

The system of partial differential equations being reduced
to the homogeneous form, Chap. xxiv. Art. 6, let n be the

number of the equations, a^, a*8, xn^r the independent

variables, and P the dependent variable.

Then from the n given equations determining

dP dP dP_

dx^ dx^ ' " dxm*

we obtain an equivalent system of equations which, by trans-
position of its terms to one side, assumes the reduced form

dP dP dP dP

-r- + An-r — + ^-7 — +^,r 7 = 0

dP dP dP dP

- — H-421-i l-^22~/ — + ^T/ — ~ \. n\

™ + Ang-+A,g- + A,~ = 0j

Multiplying these equations by the arbitrary constants

respectively, and adding the results, we have

X — \ — X —

1 dx. * dx. ' " dxn

LA '•

(2),

76 LINEAR PARTIAL DIFFERENTIAL [CH. XXV.

a single partial differential equation which, on account of the

arbitrariness of X,,X2, Xn, is equivalent to the system

from which it was formed.

Of this equation the Lagrangean auxiliary system will be
dxl _ dxa _ dxn

fef^TM: W-

whence, eliminating X,,X2, Xn, we have the system of

ordinary differential equations

- Andxl - A2idx2 ....-Anldxn =

.... -Andx = 0

dx^r - Airdx, - A,rdxs . . . .-Anrdxn = 0

These equations being included in the previous system (3),
any integrals

u = a, v = b, w = c, &c.

of them will be integrals of it. Therefore u, v, w,... will be
values of P satisfying the partial differential equation (2).
For they will be the only values which can satisfy it inde-
pendently of Xj, X2, ...... Xn. Hence they will satisfy the

equivalent system (1), and the general integral of that system
will be

F(u, v, w>,...)=0 ................. (5),

the form of F being arbitrary.

Thus the relation of the system (4) to the system (1) is the
same as the relation of the auxiliary system oif a single linear

ART. 3.] EQUATIONS OF THE FIRST ORDER. 77

partial differential equation to that equation. And the ground
of this relation is seen to be the same in both cases. The
one form necessitates the other.

3. Instead of employing the above mode of deducing
the auxiliary system, we might employ the following which
is practically more convenient.

Since any value P which satisfies the partial differential
equations determines P=c as an integral of the ordinary sys-
tem, the latter must be consistent with tZP=0 in its de-
veloped form

dP,dP dP ,

„,. . ^. dP dP dP

5V E- £„

by means of the n given equations (1), we have
dP

(axn+l A.lldx1 M.21dx.2 j3.niaxn)

dP

J^T~ (dx^r - Alrdxt - A^dx.2 ...... - Anrdxn) = 0.

Whence, equating to 0 the respective coefficients of

dP dP dP

dx^ dx^' " dx^

we have the system (4).

In the same way we can pass from the system of ordi-
nary to that of partial differential equations. From the, equa-
tion dP = 0, in its developed form, we must eliminate a number

78 LINEAR PARTIAL DIFFERENTIAL [CH. XXV.

of differentials dxlt dx^... equal to that of the given equa-
tions, and then equate to 0 the coefficients of the remaining

differentials.

4. Lastly, the formal connexion of the two systems should
be noticed. The partial differential equations "being given in the
reduced form (1), the ordinary system may be constructed as

follows : For any differential coefficient, as -^ — ,- in any

_ cfen+i ^

column after the first, write the corresponding differential
c7.rn+1, subtract from this the sum of dxv dxz, dxn, mul-
tiplied respectively by the descending coefficients of that
column, and equate the result to 0. The system of equations
thus successively formed will be the auxiliary system sought.

The transition from the ordinary to the partial system may
be effected by the same rule, substituting only differentials
for differential coefficients.

[It appears from the manuscript that an example was to
have been supplied here.]

Up to this point the theory of systems of partial differen-
tial equations is in analogy with that of single equations.
But here a difference arises. We do not know beforehand
what number of integrals a system of ordinary differential
equations, in which the number of variables exceeds by more
than one the number of the equations, admits.

The theory which removes this difficulty will be developed
in the following sections. It will be shewn that a system of
linear partial differential equations which admits of solution
by the assigning to the dependent variable a value which
satisfies all the equations in common, must either itself satisfy
a certain condition, or be capable of being developed into a
new but equivalent system which will satisfy that condition.
It will be shewn that when that condition is satisfied, the
auxiliary system of ordinary, is capable of expression as a
system of exact differential equations determining the inte-
grals sought.

ART. 5.] EQUATIONS OF THE FIRST ORDER. 79

It -will be found convenient to express by a single symbol
the aggregate of the operations to which the dependent vari-
able is subject in the expression of a partial differential equa-
tion. Thus the equation

dz dz dz

di+xd*+ydy=Q

may be expressed in the form

if we assume

= —- —

dt dx

A = —4- —4- —

'

Under this convention the following proposition is to be

understood.

5. PROP. II. If AP = 0, A'P=0 represent any two
homogeneous linear partial differential equations of the first
order, then will

(AA'-A'A)P=0

also be a homogeneous linear partial differential equation of
the first order, and it will be satisfied by all the common
integrals of the equations from which it is derived.

First, the equation will be linear. For, let x, y represent
any two variables whatever, or the same variable repeated,
out of the set xl — xn, and let A, B represent any functions
of the variables xl xn. Then A may be represented by a

series of terms of the form A -r- , and A' by a series of terms

ax J

of the form B -j- . Hence (AAr — A' A) P can be expressed
by a series of terms of the form

, d

80 LINEAR PARTIAL DIFFERENTIAL [CH. XXV.

which, on effecting the differentiations, becomes

A—^-Bd~ ~
dx dy dy dx '

the terras containing the second differential coefficients of P
mutually destroying each other. Hence the equation

(AA'-A'A)P=0

will "be a homogeneous linear partial differential equation of
the first order.

The constitution of the coefficients of this equation is easily
determined. For suppose the given equations to be

dP . dP . dP _

-a-, ~~, ~r -"•» ~i ...... «" -"n ~j — V)

1 cfcC 2 dx dx

, d A»T>
A = £-...... + ^, A-

then the equation

(AA'-A'A)P=0

may be written in the form

and, since terms involving second differential coefficients of
P will disappear, this becomes

ART. 6.] EQUATIONS OF THE FIRST ORDER. 81

We see from this that the form of the, result is the same as
if the A or A' from either equation operated only on the coeffi-
cients in the other equation.

Secondly, the above equation will be satisfied by all the
common integrals of the equations from which it is derived.

For, let <£ = c be a common integral of

AP=0 and AT =0,
then

Performing on these the respective operations A' and A,
operations which involve only differentiation together with
algebraic processes, we have

A'A<£ = 0, AA'<£ = 0,

whence, by subtraction,

AA'<£ - A'A<£ = 0,
or (AA' - A'A) <j> = 0,

from which it appears that (f> is also an integral of the
equation

(AA'-A'A)P=0,

as was to be shewn.

6. PROP. III. If by the above processes of reduction and
derivation we convert a system of partial differential equa-
tions into a new system, such that if expressed in the form

A1P=0, A2P=0, ...... AmP=0,

the condition

shall for each pair of equations be identically satisfied, then
the system of ordinary differential equations corresponding
to this new system will admit of reduction to the form of
exact differential equations, the integration of which will
enable us to construct the general value of P satisfying the
system given.

R. D. E. II. 6

82 LINEAR PARTIAL DIFFERENTIAL [CH. XXV.

1st. Suppose the given system of n equations reduced to
the form (1), marked by the peculiarity that n of the differen-
tial coefficients appear only in successive equations and with
the coefficient unity. Then taking any two of those equa-
tions (we select the first two), we have

A ^ A ^ A ^

l~ dx* ud^~ ...... + lrd^~'

ajc\ aXn+i axn+r

from the forms of which we see that the derived equation

(AA-AA)J> = o

cannot contain either

— or —

dxi dx%

It can only, as appears from Art. 5, contain the differential
coefficients

dP dP

and must be of the form

dP dP dP

~ + ~

It cannot therefore be an algebraic consequence of any of the
equations of the system (1) from which it was derived. It is,
unless by the vanishing of JSl, ....Br it present itself as an
identity, a new equation algebraically independent. Com-
bining this with the former ones, we have a system of n + 1
equations admitting of the same reduction as to form fol-
lowed by the same subsequent process of derivation. And
the result of each of these completed steps is to convert the
system into one containing one equation more than before ;
but containing in each of its equations one term fewer than
before. The process must then end either in the genesis of a
system of partial differential equations such that the further

ART. 6.] EQUATIONS OF THE FIRST ORDER. 83

application of the process of derivation of Prop. IT. shall only
lead to identities, or in the emerging of the system

-

-; — — U,

dx

The latter supposition would imply that P is a constant.
The consequences of the former we proceed to examine.

The final system of linear partial differential equations
will be of the same type (1) as the original system, but will
differ from that system in that n will be increased, and r
diminished by the same amount. We shall therefore simply
state the form (1), only under the condition

and with the altered values of n and r.

First, then, the common integrals of the new system will
be the same as those of the original system. This is evident
from Prop. II.

Secondly. If we write

m = n + r,

the first equation of the system (1) will be

and the auxiliary Lagrangean system of this will have m — 1
independent integrals

among which the n — I known integrals (Chap. XXIY. Art. 5)

*2=C2> *3 = C3> Xn = CH

6—2

84 LINEAR PARTIAL DIFFERENTIAL [CH. XXV.

are included. And the general value of P satisfying the
above first equation will be

P=F(ul,u3....um_1}.

The assumption P=x1 would not satisfy the said equation,
for it would lead, on substitution, to 1=0. Hence we infer

that while the functions MI} u2, wm_, are independent with

respect to each other, they are also independent with respect
to ajj, so that the m functions ult u2, wm_1, xl, are mu-
tually independent in the sense explained in Chap. xxiv.

Let us now transform the equations of the system (1) after
the first by introducing ult u2, ..... um_:, xt as independent
variables. Those equations being

A2P=0, ...... AnP=0,

the result of the transformation will be (Chap. xxiv. Art. 7)
dP .dP . dP .. .dP

ar

dP .dP .. . dP .. .dP

But P = xl being an integral of each of the equations of
the system (1) except the first, as appears from their forms,
we have

thus the last terms in the transformed system vanish. Further,
the coefficients of the remaining terms reduce to functions of
MI} w8, ...... wm-1 merely. For, considering the coefficient A2w,,

we have

ART. 6.] EQUATIONS OF THE FIRST ORDER. 85

which, since ^lul = 0 reduces to

Hence &sul must be a solution of AjP= 0, and therefore a
function of «,, ut, ...... um_1. And so for the others. It

results therefore that the transformed system is

.dP .. .dP . dP

«,, ua, ..... um_1 being the actual independent variables of the
system.

But the transformation having involved no loss of gene-
rality, for a new system of ra independent variables was
simply substituted for an old one, the condition

satisfied before, will continue to be satisfied in the new sys-
tem represented symbolically in the form

A,P=0, ASP=0, ...... ABP=0.

Any common integrals of this system will also be common
integrals of the previous system. For as functions of

they will satisfy the first equation of that system, and they
will satisfy the other equations, because the present system is
but a transformation of those. The converse is equally mani-
fest

Thus a system of n partial differential equations contain-
ing m independent variables and satisfying a certain condi-

86 LINEAR PARTIAL DIFFERENTIAL [CH. XXV.

tion, has in virtue of that condition been converted into a
system of n — 1 equations between m — 1 independent vari-
ables, and satisfying the same condition. This then is con-
vertible into a similarly constituted system of n — 2 equations
containing m — 2 independent variables, and so on till we
arrive at a final single partial differential equation containing
m — n + 1 independent variables. This equation has m — n,
that is, r integrals, and these are the common integrals of the
system (1).

But the system of ordinary differential equations corre-
sponding to (1) is in number r, and is satisfied by all the
common integrals of that system. Hence these differential
equations must admit of reduction to the exact form.

7. We may deduce from the above investigation the fol-
lowing Rule.

To integrate a system of simultaneous linear partial diffe-
rential equations of the first order.

EULE. Reduce the equations to the homogeneous form
(1), express the result symbolically by

A,P = 0, A2P=0, ...... AnP = 0,

and examine whether the condition

is identically satisfied for every pair of equations of the sys-
tem. If it be so, the equations of the auxiliary system,
Prop. I., will be reducible to the exact form, and their inte-
grals being

u = a, v = b, w = c, ......

the complete value of P will be F(u, v, w, ...), the form of F
being arbitrary.

If the condition be not identically satisfied, its application
will give rise to one or more new partial differential equa-
tions. Combine any one of these with the previous reduced

ART. 7.] EQUATIONS OF THE FIRST ORDER. 87

system, and again reduce in the same way. With the new
reduced system proceed as before, and continue this method
of reduction and derivation until either a system of partial
differential equations arises between every two of which the
above condition is identically satisfied, or, which is the only
possible alternative, the system •

dP dP

j-=0> -r- = 0, ...
dx^ dxa

appears. In the former case the system of ordinary equations
corresponding to the final system of partial differential equa-
tions will admit of reduction to the exact form, and the gene-
ral value of P will emerge from their integrals as above. In
the latter case the given system can only be satisfied by sup-
posing P a constant.

Ultimately then the determination of P depends on the
solution of a system of ordinary differential equations reduci-
ble to* the exact form. This does not mean that each equation
of the system is reducible to the exact form, but that the
equations may be combined together so as to form an equal
number of equivalent equations of the exact form. Generally
when we know this combination to be possible it is easy
to effect it, and best to endeavour to do so. We might how-
ever employ the method of the variation of parameters as fol-
lows. Supposing p the number of differential equations make
all but p + 1 of the variables constant, integrate the reduced
system, and then seek to satisfy the unreduced system by the
same series of integrals with the arbitrary constants as new
variables. The successive integrations and transformations
of this method would amount to the same thing as those
upon which the second part of the demonstration of Prop. in.
rests*.

Lastly, given a system of ordinary differential equations
containing a superfluous number of variables without know-
ing how many integrals they admit, we must, supposing
P = c to be any integral, construct the corresponding system

* It was thus indeed that the author was first led to that theory.

88 LINEAR PARTIAL DIFFERENTIAL [CH. XXV.

of homogeneous partial differential equations satisfied by P,
and apply to them the foregoing Rule.

8. Ex. Required the integrals of the simultaneous par-
tial differential equations

dP .dP . dP

_ + (a^ + y_^)_+(^_y)_=().

Representing these in the form A,P=0, A2P = 0, it will
be found that the equation

becomes, after rejecting an algebraic factor,

3P.*P_

X dz + dt ~

and the three equations prepared in the manner explained in
the Rule will be found to be

dP

TZ=^

dP

T. -r- = 0.
at dz

No other equations are derivable from these. We conclude
that there is but one final integral.

To obtain it, eliminate

dP dP dP

dx' ~dy ' dt

ART. 8.] EQUATIONS OF THE FIRST ORDER.

from the above system combined with

dP , dP, dP, dP, n
-T-dx+-j-dy + -:rdz + --r-dt = (),
dx dy ' dz at

dP
and equate to 0 the coefficient of -y- in the result We

find

dz-(t + 3x*) dx - ydy - xdt = 0,

the integral of which is

An arbitrary function of the first member of this equa-
tion is the general value of P.

[It appears from the manuscript that another example was
to have been added here.]

( 90 ) [CH. XXVI.

CHAPTER XXVI.

HOMOGENEOUS SYSTEMS OF LINEAR PAETIAL DIFFERENTIAL
EQUATIONS.'

1. THE theory of homogeneous systems of linear partial
differential equations in which when expressed in the sym-
bolic form

A1P=0, A2P = 0, AmP = 0 (1),

the condition

(AA-AA)^=0 (2)

is for all combinations represented by i and j satisfied in
virtue of the constitution of the symbols Af, A;-, forms the
subject of important researches by Jacobi (Nova Methodus...
Crelle's Journal, Vol. LX. p. 1). The following are the most
important of his results.

1st. An integral of any one equation of the system being
found, other integrals of the same system may be obtained
without integration, by a process of derivation founded upon
the condition (2).

Let <j> be an integral of the first equation of the system.
Then is the equation

A^ = 0
identically satisfied.

Also the condition (2) being satisfied in virtue of the con-
stitution of the symbols, we have

ART. 1.] HOMOGENEOUS SYSTEMS &C. 91

and in particular, making i=l, and separating the terms,

which reduces by a prior equation to

It appears from this that A/£, if it do not reduce to a con-
stant, is an integral of the first equation A^ = 0, and, if it
prove to be not a mere function of <f>, a new integral.

This process may be repeated upon the new integral with
a similar alternation of results. It will be evident from this
that if we confine our attention to the two equations

A1P=0, A2P=0,

and suppose, as before, <j> to be an integral of the first, then
will

or, as these may be expressed,

be also integrals of the first equation; and this process of
derivation may be continued until we arrive at an integral
A/c^ which is not independent, but is expressible as a func-
tion of prior integrals

and, sooner or later, such a result must present itself, since
the number of independent integrals is finite.

It is further seen that the most general symbolic form of an
interal derivable from the root interal <> is

a, /?, ...... ft, being positive integers.

The above remarkable theorem was in some degree antici-
pated by the researches of Poisson.

92 HOMOGENEOUS SYSTEMS OF LINEAR [CH. XXVI.

2ndly. Jacob! shews how by the aid of such derived in-
tegrals of the first equation of the system a common integral
of the first and second equation may be found, and how from
this integral and its derived series a common integral of the
first three equations of the system may be found, and so on,
until a common integral of the entire system has been as it
were built up out of previous integrals of less general appli-
cation.

Let <£, <£', <£", ...... ^-^ represent a series of independent

integrals of the equation A1P = 0, of which </> is the root in-
tegral, and the rest are derived from it by successive applica-
tions of the operation denoted by A2 , so that

also let A/<£ be not a new integral but a function of

Now <f>, <£', ...... (f>^~^ being particular integrals of AjP=0,

the function F(<f>,<j>', ...... ^~1') will also be an integral ^ of

the same equation irrespectively of its form. Let us inquire
whether the form of the function can be so determined as
to render it also an integral of the second equation AaP= 0.

We have then to satisfy the equation

By the principles of the Differential Calculus this equation
assumes the form

But

lastly, A.,^-1* may by hypothesis be expressed in the form
', ...... ^-a)). Thus the equation to be satisfied is

ART. 1.] PARTIAL DIFFERENTIAL EQUATIONS. 93

dF

a linear partial differential equation of which the auxiliary
system is

..

-»

Now the integration of this system may be made to depend
upon that of an ordinary differential equation of the (p — I)*
degree between the two variables <f>:*~1} and <j>.

For we have

d<f> $

Differentiating the last equation with respect to <f>, and attend-

di<f><i'-V .

ing to the former ones, we shall be able to express J in

terms of the variables <f>, <f>, ...... ^-1). Proceeding with

this in the same way and continuing the process we shall be
able to express the series of differential coefficients

'

in terms of <£, <f>, ..... ty*'
eliminating <£', ^>", ..... ^>'M~^, we
between

Lt Y/

From these p, — 1 equations,
'G shall have a final equation

' d<j> '" dp-L '

that is, a differential equation of the (jj, — 1)°* order between
and $<»-».

94 HOMOGENEOUS SYSTEMS OF LINEAR |_CH.

The complete integral of this equation will be of the form

Differentiating this //, — 2 times in succession with respect to
<b, and continually substituting for the differential coefficients
of (^"^ their values as before assigned in terms of

we shall have a system of /* — 1 equations connecting the
above variables with the constants c1? c2, ..... cM_i. Finally,
solving these equations with respect to the constants, we shall
possess the integrals required in the form

and each of these will be a common integral of the first two
equations of the given system (1).

[On the back of a page of the manuscript the following
paragraph occurs, which seems to have been intended as a
simplification of the preceding argument which begins with
"The complete integral."]

Suppose that a first integral of the equation can be found.
Its form will be

Substitute in this for the differential coefficients of <f>(t*-l>
their values before assigned in terms of </>, </>', (f>",...(f>lp~l\ and
we have an integral of the system (3), and therefore a com-
mon integral of the first two equations of the system (1).

[We now return to the place at which we inserted a para-
graph.]

Just in the same way Jacobi deduces a common integral of
the first three equations of the system (1). For representing

ART. 2.] PARTIAL DIFFERENTIAL EQUATIONS. 95

any one of the first members of the above system by i/r. and
deriving thence the new independent integrals Asi/r, A^^r,...
he substitutes an arbitrary function of these for P in the
equation

ASP=0.

It is evident that the solution of the partial differential
equation so found will again be reducible to that of an
ordinary differential equation between two variables. And
so the process is carried on till all the equations are satis-
fied.

2. The above remarkable process was developed by Jacob!
in connexion with the theory of non-linear partial differential
equations of the first order. In that particular connexion it
admits of certain reductions tending to diminish the order of
the differential equations to be integrated. But these do not
affect the general principle of the method. It was in this
special form that the theory of the solution of simultaneous
linear partial differential equations originated. Jacobi does
not consider the theory of equations in which the condition
(2) is not satisfied ; but the language in which he refers to
the condition shews that he had speculated upon the general
problem — and it is difficult to conceive that he should have
meditated upon it and not arrived at its complete solution.

[The manuscript here gives the first two words of
the passage from Jacobi's memoir which is quoted in the
Philosophical Transactions for 1863, page 486.]

(96 ) [CH. xxvn.

CHAPTER XXVII.

OF NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS OF THE
FIRST ORDER.

1. IN treating the present subject we shall first consider
that class of non-linear partial differential equations of the
first order which involves two independent variables, and
then proceed to the general theory. The reason for this
procedure is that the particular theory, though of course in-
cluded in the general one, rests upon a somewhat simpler
basis, and it was in fact developed by the labours of
Lagrange and Charpit long before the general theory was
known. The latter we owe to the independent researches
of Cauchy and Jacobi.

[Here the manuscript refers to the matter contained in
Chap. xiv. Arts. 7 to 12 inclusive; and then passes on to
the general theory.]

General Theory.
2. Given an equation of the form

the number of arbitrary constants a,, a2, ... an involved being
equal to the number of the independent variables xv xz, ... XM
we obtain by differentiation and elimination of the constants
a partial differential equation of the first order. Of this the
proposed equation is said to constitute a complete primitive.

ART. 2.] XOX-LINEAR PARTIAL DIFF. EQUATIONS &C. 97

The form of the above process which it seems best, as
throwing light upon the inverse problem of deducing the
complete primitive from the partial differential equation, to
employ, is the following. Let the given primitive, solved
witli respect to one of the arbitrary constants av be presented
in the form

f(xlt ... xn, z, oa> ... aj =a^ ...... (1).

Differentiating with respect to each of the independent vari-
ables we have a system of n equations of the forms

(2).

These n equations enable us first to eliminate the n — 1
constants a^, ...... an, and so deduce the partial differential

equation sought in the form

^fo,... x*,z,2>l,...jpj=0 ............ (3);

secondly to determine the n — 1 constants as functions of
xlt ... #n, z,p^, ... pn in the forms

(xlt ... xMe,plt ... /)») =aa]

...........................

(xlt ... XH, z,plt ...pn)=anj

As the system formed of these n — 1 equations, together
with the previous one, is merely another form of the system
(2) obtained by directly differentiating the primitive, it follows
that if from these equations we deduce the values of^, ...^m
as functions of xlt ...xH,a3, ...an, and substitute them in the
equation

they will render that equation integrable, and its- integral
will be the complete primitive (1), the constant a^ being re-
gained by integration.

B.D.E. IT. 7

98 NON -LINEAR PARTIAL DIFFERENTIAL [CH. XXVII.

Examining the system (3), (4) we see that the first mem-
bers of all the equations which it contains are functions of
xl, ... xn, 2, jp1? ... pn, while the second members are con-
stants. The question then arises, What mutual connexion
exists among these functions in virtue of which they yield
values of plt ...pn, which render the equation (5) inte-
grable?

The answer to this question must involve the entire theory
of the solution of partial differential equations of the first
order, so far as relates to the determination of a complete
primitive. Given a partial differential equation of the form
(3) it is evident that if we can construct a system of associated
equations (4) possessing the character above described, the
final value of z obtained by integration of (5) will both
satisfy the given equation and contain the requisite number
of arbitrary constants. It does not follow from this that
it will be the only complete primitive, but it will be a
complete primitive.

3. The relation sought is expressed in the following
Proposition :

PROPOSITION. If

<£(#,, ... xn, ztplt ...pn)=b

represent any two out of a system of n independent equations
such that the values ofp^ ...pn, thence determined would make
the equation

dz =d

integrable, then the first members of these equations lei/iff
represented for simplicity by F and 3>, the condition

F\d$> dF/d®

the summation extending to all values of i, from 1 to n inclusive,
will be satisfied identically.

ART.

EQUATIONS OF THE FIRST ORDEE.

99

Reciprocally, if the above condition be satisfied identically
for each binary combination of functions in the proposed system
of equations, and if these functions be independent, then the
values ofpl .... p^as functions ofx^ ...xn, z, which they yield,
tcill make the equation

dz =

integrdble.

It will be convenient to begin with the particular case in
which the proposed equations do not explicitly contain z, the
particular pair to be considered being represented by

Differentiating with respect to ar£, and regarding^, ...pn as
functions of the independent variables, we have

dxi

d® ^i+ +^^=0 I

(6),

which we may give the form

*E - - 1 — dp/'
dxt ' dp* dx\

d® _ ^ d® dp}-

(7),

ie summation with respect toy extending fromy= 1 to j = n
iclusive.

From the first of equations (7) multiplied by -5- subtract

7—2

100 NON-LINEAR PARTIAL DIFFERENTIAL [CH. XXVII.

- rJW
the second multiplied by -j— , and sum the result with respect

to i from i= 1 to i = n inclusive. We have

2 ldF_ d<& _d_F d3>\
"l \f[xt dpi dpi dxj

___ ( ,

'* 3 \dpj dpi dxi dpi dpj dx

The expression under the double sign of summation in the
second member vanishes when *=,/; we may therefore re-
strict the summation to unequal values of i and j. Now
as for any particular combination of values, e.g. 2, 3, there
would exist in the completed member both the terms cor-
responding to t = 2,J = 3, and those corresponding tot/ = 2,
i=3, it is evident that if we employ the symbol 2y to denote
summation with respect to different combinations of i andj,
the second member of the last equation may be expressed in
the form

^ fdF d<£> dpj _ dF d3> dp,.
v \dj)j dpt dxi dpt dpj dxi

dF^d^dpi_dFd^
dpi dpj dxj dp3- dpi

^ (fdF d$> dF d®\ fdpi dp,

Q-M > J / _ _ . _ _ j I _-*. mm _ _*_JL

lj \\dpi dp; dpj dpi/^dXj dx

so that the equation (8) becomes

dF^
dpi

_^ (fdJ^d^_Wd^\(d^_dpJ\\
^ \\dfr dPi dPj dpj \dx}- dxj}"

The number of terms of which the second member ex-

M ( ft ^_- 1 j

presses the sum is thus - - , and it will be observed that

ART. 3.] EQUATIONS OF THE FIRST ORDER. 101

as to any particular term it makes no difference in what order
the numerical values of i and j are assigned to these quan-
tities; e.g. whether for the combination 2, 3 we make t = 2,
j = 3, or i = 3, j = 2; but we must confine ourselves to one
order.

Now when the equation
dz =

is integrable in the manner here supposed, we have for all
combinations of i andj,

All the terms in the second member of (9) therefore vanish,
and we have

2 fdF_ d®_d]^ d®\
"* \dxi dpi dpi dxj

This is the direct form of the Proposition under the parti-
cular limitation supposed.

As F, 3> represent, under the same limitation, any two of
the first members of the n equations (3), (4), which determine

pl} '-.pn, there will exist —~- — - equations like the above.

21

It is usual to employ for brevity the notation

2 (dF_ d^_dF d^\_
i(dxi dp, dp, dxj~l j>

and this being done the above system of equations expresses
the - - - functions of the form [-^^ as linear homogeneous

s S .1 n (n ~ 1) , «, • j? -L _e dpi dp,

functions of the - - quantities of the form -~- — -/- .

& dXj dx\

It is hence that the vanishing of the latter series of quantities
secures the vanishing of the former.

The converse truth will therefore be established by shewing
that the — ^— — - quantities of the form -~ — -f1 are. when

102

NON-LINEAR PARTIAL DIFFERENTIAL [CH. XXVII.

Flt Fz , . . . Fn are independent with respect to p^ p2, ...pn,
expressible as linear homogeneous functions of the — =-?

9

functions

To avoid complexity of expression I shall establish this for
the particular case of n = 3, and shall shew that the reasoning
is general.

The functions Flt F^ F3, being independent with respect
z the determinant

df

dF

<*&'

does not vanish. This determinant we shall denote by A.

In (9) writing for .Pand <3> first F2 and F3, secondly F3 and
Flt thirdly Ft and Fa, we have on changing signs the system

(10).

dp 3 dp2J \dxa dxf^
dp3 dpz ~dpj \dxs dxj

Multiply the first equation by -j— • , the second by -, 2, the

third by -j- - and add. Then
dp,

dp,

dp, *'

ART. 3.] EQUATIONS OF THE FIRST OBDEE. 103

whence as A does not vanish we have, on dividing by it, the

function / -- — ~ expressed as a linear homogeneous function
dxs dxz

of [F^, [F.F'l, and

In like manner multiply ing the equations by -^ , -=— - , -y-1

J dPi dp% dp,

respectively, and dividing by A, we obtain -j- — -^-l as a

CLX - d3ts~

similar linear homogeneous function, and lastly, multiplying

dF c!F

,
by

-7— -, -j—,

,
and proceeding as before, we obtain

jp — -?* as a similar linear homogeneous function.
dxa dx1

From all which it follows that when [J^FJ, [F3FJ,
vanish, then

_ fa dj>^_ dp^

dx.

dx.

will vanish also.

The reasoning is general in its nature. If Ft. F2, ...F
are independent with regard topltptt ...pn, the deteriuiuant

= A

loes not vanish. This determinant is from its constitution
a determinant linear and homogeneous, not only with
spect to any row or column of elements, but also with
ispect to the possible binary combinations which can be
formed of two rows or columns, ternary out of three rows or
)lumns, &c. provided that these combinations are themselves
}f the form of determinants. In the language of the theory
uch combinations are called minor determinants. Hence if
re construct the system of equations represented by (10), and

104 NON-LINEAR PARTIAL DIFFERENTIAL [CH. XXVII.

observe that the coefficients of any particular term of the form

•*-*• — ^ in the several equations form a system of such
dxj dxi

minors to the general determinant (11), it will be plain that
the equations can by multiplication and addition be brought
to a form in which the coefficient of that particular term will
be A. At the same time the coefficients of all the other

terms of the form -~ — §p will vanish. For a little atten-
dXj dXi

tion will shew that they will be what the determinant A
would become on making two of its columns or rows of ele-
ments equal, and therefore will be identically equal to 0.

Thus the Proposition is generally established for the case
in which z does not explicitly appear in the functions

F F F
-*• i j * g i ..... -*• » •

When z does appear in those functions the equations (6)
will be replaced by

. ., ..

dxt "* dz dpt dxi dpn dxj,

d<£> d<& J<E> dpl d<& dpn _

dxi ** dz dpl dxi dpn dx^

from which it is seen that the theorem above established will
only need to be changed into the form employed in the state-
ment of the general Proposition.

As the above is one of the most important propositions in
the entire theory of Differential Equations, it may be desire-
able to illustrate it by examples.

[There are no examples in the manuscript.]

4. We resume the general theory.

The integration of non-linear partial differential equations
may be effected by two distinct methods, both resting upon

ART. 4.] EQUATIONS OF THE FIRST ORDER. 105

the ground of the above Proposition. The first of these
methods, originally established by a different analysis from
that which will here be employed, was discovered by Cauchy
(Exercices d' Analyse), and rediscovered by Jacobi (Crelles
Journal). The second method, discovered by Jacobi at a
later period, forms the subject of his posthumous memoir,
Nova Methodus...

Cauchy s Method.

We will, as before, begin with the case in which z does
not appear explicitly in the proposed partial differential equa-
tion, which we shall represent in the form

•Fi(*i> ..... *., plt ..... lO =0 .............. (1).

We have seen that to find a complete primitive of the
equation it is necessary and sufficient to construct a series of
equations

F*(xv ..... *-» Pi. ..... /0=«tl

............................... ............ (2),

F»(*i ..... ar.»l»i» ..... |0=*«J

such that not only shall the conditions

connecting the new functions F3, ..... FH with Flt be identi-
cally satisfied, but also the series of conditions

GO,

Fa and Ft representing any two of the new functions re-
ferred to.

The first of the above series of conditions amounts to
this, that jP2, ..... Fn must be integrals of the partial differen-
tial equation

o ......................... (5).

106 NON-LINEAR PARTIAL DIFFERENTIAL [CH. XXVII.

It is the peculiar aim of Cauchy's method to determine the
integrals so as to cause the second series of conditions to be
satisfied also. And it is shewn that this will be attained if
the integrals of (5), which form the first members of (2), are
such that the particular values which p^ ...._pn_1 assume when
xn is made to receive any constant value, as 0, are differential

coefficients with respect to a^, xn^ of any single function

of those variables, the form of which may be arbitrarily
assigned.

The necessity of this condition is obvious. If the general

values of p^ pn are differential coefficients of a function z

with respect to xlt xn, then the particular forms which

j? , p i assume when xn receives any constant value are

simply differential coefficients with respect to x, xn_1 of

what, z becomes under the same circumstances. To prove its
sufficiency we must shew that when it is satisfied the condi-
tions represented by (4) will be satisfied also.

Since Fa and Fb are integrals of [FJP] = 0,

= 0.. ,..(6).

Also, since if in (t) and (2) we give to xn a particular con
stant value, as 0, and then in (2) regard pn as a function of

determined by (1), the system (2) will virtually contain only

X\1 ..... Xn~U Pl> ..... Pn-\1

of which jt?j, ..... pn_1 are differential coefficients of a single
function with respect to x^ ..... xn_^ it follows from the pro-
position of Art. 3, that any two functions Fa and Fb will
satisfy mutually the condition

2i«.-i (dF* d]\ _ dFa dF\\ =. Q
i-1 \dxi dpi dpi dxj

the differentiations having reference to

ART. -4.] EQUATIONS OF THE FIRST ORDER. 107

explicitly as they appear in Fa and Fb, and implicitly as
involved in p». Thus the developed form of the above equa-
tion is

i=n_! (/dF_a d_F» dfa ftF, dFb dp\
iU* + dpn dxj (<fr + dpn dpj

_ ,
(dp, + dpn dpj (dxt + dpu dx ~

the forms of ~ and -f^ beinor determined from (1).

Performing the multiplications, the above equations will be
reduced to the form

_
dpi

_
dpn \dxt dpi dpi

dj\ ^=n_, (dpn dFa dpn dF«\ _
dpn2i=l (dx, dp, fa dxi)-

But from the form of the total differential of (1) we see that

df\ df\

dpn _ dxj dpa _
=~

Hence

dp,, dFa

i=.-i fPn d_Fa _ dp,, dFa\
(dxi dpi dp, dxj

(Wr 2f --i (dK dF. _ dj\ dFa\
\dpj : \dXi dpi dpi dxj

108 NON-LINEAR PARTIAL DIFFERENTIAL [CH. XXVII.

Now since by Art. 3

^=nfdF\dJ\_dF\dF\\=Q
1 1 \dxi dpi dp i dxj

we have

i dpi dpi dx

_

\dxn dpn dpn dx

therefore

_
t-1 \dxi dpi dpi dxj

= (^}~l (dl± ^ _ ^5 d_

\dpj \dxn dpn dpn dx
In the same way

^i-.-i/^ ^_dP^
"~t~1 \clxi dpi dpi dx

_
\dpj \dxn dpn dpn dx

The substitution of these values in (7) gives

^n-i(dladF,_dF. dF>\
~i=1 \dxi dpi dpi dxj

fdFY-1 (dF, fdf\ dFb _ dj\ dFb\
\dpj \dpn \dxn dpn dpn dxj

^_dF, fdF\ dF,>_dFt df].\\_0
<*P* \da* dpn dpn dxj] ~ '

ART. 4.] EQUATIONS OF THE FIRST ORDER. 109

\dxi dp i dpi dxj

_ =

dxn dpn dpn dxn

^n(dF,dF*_dF_adJ\\ =
Zi=1U*i dfi dPi dxj

which is precisely the equation

We see therefore that to solve the partial differential
equation

FI(*> ..... *„, P*, ..... jO=0»

it is only necessary to construct the linear partial differential
equation

^n__

Zi=1Uri dpi dpi dxj~

and to obtain n — 1 independent integrals of this

Fn(x^ ..... xn, plt ..... ?„)=<*„,

such that if we determine from these conjoined with the given
equation the values of pl,....pn, then those ofj91? ..... p
shall, when xn is made constant, be the partial differential
coefficients of one and the same function of xl} ..... x^ with
respect to these variables in succession.

Now provided that we can find all the integrals of the
above partial differential equation the particular determination
required may be effected in the following manner.

The Lagrangean auxiliary system consists of 2n — 1 ordi-
nary differential equations

dp* dx^ dxn

1.10 NON-LINEAK PARTIAL DIFFERENTIAL [CH. XXVII.

These admit of 2n — I integrals, one of which will be F1 = cl;
and this will agree with the given equation if we make ct = 0.
We have therefore, besides the particular integral Fv = 0,

2n — 2 integrals of the form

Now suppose it required to find a value of z as a function
of xl , . . . xn , which shall satisfy the given partial differential
equation, and shall reduce when xn = 0 (or any numerical con-
stant.) to a particular given function of cc,, ...#„_!, which we
will represent by •x/r (a^, ... a^J. Then on the assumption
that xn = 0, we have first the given equation

z = Tjr(x1,...a:n_l) ..................... (10),

secondly the derived equations

_

_<ty (*,,... «?,_,)

(11).

Make in the 2n — 1 integrals xn = 0, and suppose at the
same time o^, ... xn_l, p^ • ••JV,_i to receive therein the same
values as in the above derived equations. Then from the
3n — 2 particular equations which we thus possess in the two
systems united (particular because under the assumption that
xn = 0), we can eliminate the 2n—l particular values of
£Cj, ...#„_,, p ...pn_^ and so obtain n — I equations among the
constants. These express the conditions which are necessary
and sufficient in order that the values of pl , . . . pn^ thus derived
from the integral equations may, when xn = 0, agree with the
values assigned in (11). Accordingly if we substitute in these
equations of condition for c2, . . .c2n_^ the general values </>2 , . . . fa^
we shall obtain n — 1 equations between xlt ... #„, pl, ... pn,
which will at once be particular integrals of the system (8),

ART. 4.] EQUATIONS OF THE FIRST ORDER. Ill

and will possess the property that the values of pl , . ..pn^ which
they in conjunction with Fl = 0 give will when xn = 0 reduce
to the values given in (11). Hence these values with that of
pn derived from the same equation will make

dz -Pl dx^ - ..... -pn dxn = 0

an exact differential equation. In the integral of this it will
only remain to determine the constant so as to make the value
of z agree with that given in (10). All the conditions will
then be satisfied.

We may collect the results of the above investigation into
the following Eule :

To obtain an expression for z as a function of the inde-
pendent variables x^ ...xn, which shall satisfy the partial
differential equation

and shall when xn is made equal to 0 (or to any numerical
value) reduce to a given function of x1} ... x^_1} which we will
represent by tyfa, ...o^J.

RULE. Construct the linear partial differential equation

^n_ =

Zi=l\dxi dp, dp, dxj

and forming its auxiliary Lagrangean system deduce its in-
tegrals

<£2 = C2» • • • </>2»_l = C2n_l>

in addition to the known particular integral F= 0.
Between the above integrals and the equations

eliminate, after making a\ = 0, the quantities

112 NON-LINEAR PARTIAL DIFFERENTIAL [CH. XXVII.

In the resulting n — 1 equations replace

C2b7 &> • "<W by </>,„_! ,

and we shall have a system of equations which with F=Q
will determine values of^, ...^?n, which will render

dz -p1dxl - ...-pndxn

an exact differential. The integration of this will give the
integral sought.

In the case in which the given partial differential equation
is of the form

F(xlt ...xnt z, plt ...j?J = 0,

z being contained explicitly, the linear equation to be solved is
dF dF\dP dFfdP

and the argument by which it is shewn that the integrals of
this to be employed in conjunction with F=0 for the deter-
mination of •pl , ... pn need only be so conditioned as to make
p1, ...^n_i differential coefficients of one and the same function
of #j , . . . aVi when xn = 0 is in character the same as that
already developed in the present Article. It is only necessary

. . dF dF , dF
to substitute in its exposition -, — \-pi -7- tor ^— , and so

for the other functions.

But as the auxiliary system

dx dxn dz

_dF~ _ dF_ _ dF
dp, dp. &dPl Pndpn

~ dF dF dF^ dF"

dx^Pl dz dxn+Pndz

virtually includes the equation

dz -pidx^ - ... -pndxn = 0,
the ultimate expression of the Rule will be as follows :

ART. 5.] EQUATIONS OF THE FIRST ORDER. 113

To obtain an expression for z as a function of a^, ... xn
which shall satisfy the equation

... x

and shall when xn is made equal to 0 (or to any particular
constant value) reduce to a given function ty (x1} ... a^) of
the independent variables x1} ... x^.

RULE. Let

be the 2n — I integrals of the auxiliary system (12) which are
additional to the particular integral F= 0. Make in these 2n
equations arn = 0 and forming the further equations

z = -f (xlt ... or,.,),
, ... ar,J

-- -

eliminate the 2n quantities a^, ... #„_!, z, plt ... pn.
obtain n equations among the constants c^,...^.

Substitute in these equations <f>3 for c2, ... $,„ for Cj*, and
we have n equations connecting #,, ... #„, z, _p17... ^?», from
which with the aid of the given equation^, ... pn may be
eliminated, and there will result a single equation connecting
xl} ... o\ with z. This is the integral sought.

[It appears from the manuscript that an example was to
have been supplied here.]

5. Cauchy's method is evidently a general one. But its
generality is not of the same kind as that which belongs
to Lagrange's solution of linear partial differential equations.
It conducts us, not to a form embracing every possible

B.D. E. II. 8

114 NON-LINEAR PARTIAL DIFFERENTIAL [CH. XXVII.

solution, but to a system of results from which every possible
solution may be derived, by arbitrarily varying the form of
the function which expresses the initial state of the dependent
variable, that is the value of z when xn = 0, and then per-
forming certain eliminations. To obtain a complete primi-
tive we should only have to assume as the form of z when
xn = Q a function of the variables xv ... xn_1 involving n
independent constants. The form of this function is arbitrary.
Each distinct determination of it under the conditions leacU
to a distinct complete primitive. The number of such com-
plete primitives is infinite.

There are some most important problems in which the
knowledge of a single complete primitive is all that is re-
quired. For this purpose the method of Jacobi which we
shall now give may be employed.

Jacobi's Last Method.

6. Supposing z to be not explicitly involved in the given
partial differential equation

which we shall as before represent by Fl = 0, the problem of
the discovery of a complete primitive consists in the finding
of n — 1 equations

F = o F =a

2 U2' ............ *• — Wn>

such that between any two functions FtFj the relation

0 ......... (1)

shall be identically satisfied. The values of p^ , ... pn deduced
from the equations, by rendering

dz —pidxl — ... —pndxn= 0
integrable lead us to the complete primitive expressed by its

integral,

ART. 6.] EQUATIONS OF THE FIEST OEDEE. 115

Xow the idea upon which Jacobi's later methods rest is
that of directly solving the different systems of linear partial
differential equations flowing from the general condition (1),
not of solving, as in Cauchy's method, one of those equations
and then limiting that solution by conditions which virtually
involve the satisfaction of the others.

It is evident that the entire series of — ^ -- - conditions

(1) will be satisfied if we determine Ft to satisfy the single
equation

then F3 to satisfy the system of two simultaneous partial
differential equations

then FI to satisfy the system of three simultaneous partial
differential equations

and so on, until finally Fn is determined by the solution of
the system of n — 1 partial differential equations

n] = 0, [F2FJ = 0, ...... [J^JPJ = 0.

Now all these are particular cases of the general problem
of determining a function P which shall satisfy simultaneously
the equations

[FtP] = 0, [FJP] = 0, ...... [FnP] = 0. (2)

F^ Fa, ... Fn being given functions between each pair of
which the equation

is identically satisfied. Here P will represent in succession
the series F,, F3, ... Fn.

The given system is one of homogeneous linear partial
differential equations. It belongs to the class of systems the

8—2

116 NON-LINEAR PARTIAL DIFFERENTIAL [CH. xxvu.

general theory of which is discussed in Chap. xxvr. But it
is not necessary to apply the theory in its general form. We
need only a single integral ; for a single value of each of the
functions jP2, F3, ... Fn suffices in combination with the given
value of Z\ for the determination of a complete primitive.
Now it may be shewn that the system is of the class dis-
cussed in Chapter xxvi. If expressed symbolically in the
form

A1P=0, A2P=0,... AnP = 0,
the condition

(AA--AA)P=O,

will be identically satisfied. Hence Jacobi's method for the
treatment of -systems of this kind may be applied.

That the system is of the kind asserted is a consequence of
the following proposition.

PROPOSITION. If the equations

[«P]=0, [>P] = 0
are expressed in the symbolic form

AP=0, A'P=0,
then the derived equation

(AA'-A'A)P=0 (3),

will be equivalent to

f[ uv] P | = 0.

^i=n d du d

For A = Zf=i K— -j -- -j— -j— ,

i dpi dpi dxj

.
dp,, dpi dx '

ART. 6.] EQUATIONS OF THE FIRST ORDER. 117

Hence since - - is the coefficient of -y- in A'P, and -—

its coefficient in AP, its co efficient in the derivedequation (3;
will be (Chap. xxv. Art, 5),

. dv A , du

A -j A -T-;

dxj ax.

or

or

d*u dv_ d*u \

t=n/du d*v du d*v _d

1 \dxi dpidxj dpi dxidxf dxi dp^dxj dp^

d ^,-=n fdu dv du dv

or

In like manner the coefficient of -^— is

Hence (A* - A'A) P = ^ (*& f -^ f )

•rl \ ^c, ^ dfi <faj'

whence the Proposition is established.

Applying this to the system (2) we see th'at any derived
equation will be of the form

But [FtF~\ = 0 by the conditions given ; hence the condi-
tion (AiAy — A;A4)P = 0, is identically satisfied.

The results of Chapter xxvi. being thus directly applicable
to the system under consideration, we see that a common
integral of the system (2) may be found by a series of alter-

118 NON-LINE AE PARTIAL DIFF. EQUATIONS. [CH. XXVII.

nate processes of integration and derivation. We begin by
seeking an integral of the first partial differential equation.
By a process of derivation, always possible, followed by the
integration of a differential equation between two variables,
we arrive at a common integral of the first two partial diffe-
rential equations. Again, by a process of derivation followed
by the solution of a differential equation we obtain a common
integral of the first three partial differential equations. And
so on, until a common integral of all is obtained.

7. Another solution of the above problem has recently
been given. Beginning as in Jacobi's method by finding an
integral of the first partial differential equation, a process of
derivation agreeing in principle with Jacobi's, only more
extended, may lead us without further integration to a point
at which the discovery of a common integral of the entire
system will depend only upon the solution of a single diffe-
rential equation of the first order susceptible of being made
integrable by a factor. Failing this, it will enable us to
convert the given system of partial differential equations into
a new system possessing the same general character, but con-
taining one equation less. Upon this the same process may
be tried with a similar final alternative — and so on till the
required integral is discovered. (On the Differential Equa-
tions of Dynamics. Philosophical Transactions, 1863).

CHAPTER XXVIII.

PARTIAL DIFFERENTIAL EQUATIONS OF THE SECOND ORDER.

[THIS Chapter is a reconstruction on a larger scale of part
of Chapter xv. At the end of the Chapter reference will
be given to other writings of Professor Boole on the subject
here discussed.]

1. The general form of a partial differential equation of
the second order is

F(x,y, z,p, q, r,s, <) = 0 ............ (1),

where

dz dz d"z d*z d?z

P=^7~> ?=j~» r=j~s> s= J J > ^~T»'
* dx * ay ax ax ay a if

It is only in particular cases that the equation admits of
integration, and the most important is that in which the
differential coefficients of the second order present them-
selves only in the first degree ; the equation thus assuming
the form

Tt = V ...... ................. (2),

in which E, S, T, and V are functions of x, y, z, p and q.

The most important part of the theory of the solution of
this equation is due to Monge, and was extended by Ampere
to the more general equation

Rr+ Ss+ Tt+ U(s*-rt)=V ............. (3).

120 PARTIAL DIFFERENTIAL EQUATIONS [CH. XXVII I.

This equation, together with the particular equation of
Monge, and the equation

both which though falling under Ampere's general form
possess peculiarities demanding special notice, I propose to
consider in this Chapter. I shall in conclusion make some
observations on the theory of partial differential equations of
the second order with more than two independent variables.

Monge's method, and Ampere's in so far as it is an exten-
sion of Monge's, consists in a certain procedure for discovering
either one or two first integrals of the form

~=/(f) ............................. (4),

u and v being determinate functions of x, y, z, p, and q ; arid
f being an arbitrary functional symbol. From these first in-
tegrals, singly or in combination, the second integral involving
two arbitrary functions is obtained by a subsequent inte-
gration.

Now this procedure involves the assumption that the pro-
" posed equation admits of a first integral of the form (4). But
such is not always the case. There exist primitive equations
involving two arbitrary functions, from which by proceeding
to a second differentiation both functions may be eliminated
and an equation of the form (2) obtained, but from which it-
is impossible to eliminate one function only so as to lead to an
intermediate equation of the form (4). Especially this hap-
pens if the primitive involve an arbitrary function and its
derived function together. Thus the primitive

z = <f> (y + x} + ^ (y - x} - x {f (y + x) - -^ (y -a?)). ..(5),
leads to the partial differential equation of the second order

(6),

X

but not through an intermediate equation of the form (4).

It is necessary therefore, not only to consider the case in
which the assumed condition is satisfied, but also to notice

ART. 2.]

OF THE SECOND ORDER.

121

what has been done in those cases which do not at present
fall under the dominion of any known method.

Genesis of the Equation.

2. PROP. I. A partial differential equation of the first
order of the form u =f(v], or its symmetrical equivalent,

F(u,v)=0,

in which u and v are any functions of x, y, z,p, q} always
leads to a partial differential equation of the form

Er + Ss + Tt+U(s*- rt] = V.

For, differentiating the proposed first integral with respect to
x, and with respect to y, we have

dv

dq

. J>i du du du

-r- \-r + j~P + ~rr + -j-8
du \ax azc dp dq

dFfdv (fa dv_
dv \dx dz" dp

dF (du du du du

du \dy dz" dp dq

dFfolv_ dv

dv \dy dz '

dr
'dp

For brevity, write

'du

da:,

du
dx

du
>dz

\ /. l*t* Ull/ •, f'-f'-l.\ f €*£

-r- 1 lor - + p -j- , and H- for -j-
^ \J~l J

'du\

JyJ

du
fy

flu

and then eliminating

dF dF
du ' dv '

122

PARTIAL DIFFERENTIAL E

we have

(fdu\

, du

L^ol

\(dv\

dv

\(dx)'

f -j-r-
dp

r ~J~ S( '
dq j

\\d~y)'

dp

(fdu\

du

du I

i (fdv

dv
-r
dq

dv dv

r -T -j-nj- -j~r + j-
(\dyj dp dq ) (\dxj dp dy

which, on effecting the multiplication, gives

(du fdv\ fdu\ dv]
[dp \dy) \dy) dp] r

(fdu\ dv du rdv\ _ fdu\ dv du fdv\
\\dxj dp dp (dx) (dyj dq dq \dy)

(C

du\ dv du fdv^..
dx) dq dq \dx/

du dv du dv
\dq dp dp dq

_ fdu\ fdv\ /du\ fdv
\dy) \dxj \dxj \dy

a result which, since u and v are by hypothesis given func-
tions of a?, y, z, p, q, is seen to be a particular case of the
general form (3).

We may hence deduce also the conditions under which
particular forms included in the general form (3) arise. Thus,
in order that the equation u =f(v] may give rise to a par-
tial differential equation of the second order of Monge's form

it is necessary that the condition

du dv du dv _
dqr dp dp dq

should be identically satisfied. This requires, by Chap. IT.

ART. 3.] OF THE SECOND ORDER. 123

Art. 1, that u and v, considered as functions of p and q, should
not be independent.

3. The geometrical relations of the equation (3) are also
remarkable. It may in particular be shewn that an equation
of this form will be satisfied by the equation of any surface
which constitutes the envelope of any system of surfaces
formed by the variation of three parameters in subjection
to two arbitrary conditions. For let the common equation of
the enveloped surfaces be

the parameters a, b, c varying in subjection to the conditions

conditions which, determining b and c as functions of a, may
be reduced to the form

J = 0(a), c = +(a] ................... (9).

Now the values of p and q being the same for any point
in the envelope as for the same point in the generating surface,
we have for all such points

_ df(x, y, a. b. c} _ df(x, y, a, b, c)

dx d

These two equations in conjunction with (9) enable us to
determine a, b, c as functions of x, y, z, p, q. Let these
values be

a = u, b = v, c = w.

Then substituting in (9) we have

equations which hold for all such points. These are then the
partial differential equations of the first order of the envelope.

Xow each of these equations is of the general form (4) ;
whence by Prop. I. the partial differential equation of the
second order is of the form (3), as was to be proved.

124 PARTIAL DIFFERENTIAL EQUATIONS [CH. XXVIII.

Let us actually construct this equation.

Differentiating the first of the equations (10) with respect
to x and to y, and regarding therein a as a function of those
variables, and b and c as functions of a, we have

r = d'f i f d*f [ a*f db t d*f dc\da
dx" \dadx dbdx da dcdx da) dx '

S=^L+(^L d*Ld± d*f_dc\da
dxdy \dadx dbdx da dcdx da) dy '

from which we readily derive

/ d*f\ da ( d*f\ da

I n* _ ^ _ 11 1 _ __ j o _ __ *f _ j _ _ — Q

\ dx2 J dy \ dxdy) dx

Proceeding in the same way with the second equation of
the system (10) we have

_

_

dxdy dy \ dy* dx

Hence, eliminating -v- and -j- , we have

/ rfVV f d*f\(f d*_
\S~dxdy) V ~dx*

..-__

dy* dxdy dx* dx* dy* \dxdy) '

the equation sought.

Comparing this with the general form (3) we have the equa-
tions

_2 _

dy* _ dxdy _ dx* __ 1 _ dx* dy* \dxdij 1

~r"~u:~ "T"" "

ART. 4.] OF THE SECOND ORDER. 125

d'f d*f tff

whence eliminating ^«, -=i, and —-,- we amve at the
1 dxr dy dxdy

equation,

This then is the condition which must be satisfied in order
that the equation (3) may admit of an integral representing
the envelope of a system of surfaces in which three parameters
vary in subjection to two connecting conditions. It is only
proved however to be a necessary, not to be a sufficient, con-
dition.

Solution of the equation Rr -f Ss -f Tt + Z7(s* — rt) = V, when
a first integral of the form F(u, v) = 0, exists.

4. In the following sections we propose

1st. To shew that when a first integral of the above form
exists, its discovery depends upon the solution of two simul-
taneous partial differential equations of the first order re-
solvable into linear equations.

2ndly. To shew how from such first integral or integrals
the second integral is to be obtained.

PROP. II. If the equation

Mr+Ss+Tt+U(s*-rt)=V

admit of a first integral of the form F(u, v) = 0, in which u
and v are functions of x, y, z,p, q, then will F(u, v) considered
as a function ofx, y, z, p, q, and represented as such for brevity
by F satisfy the two partial differential equations of the first
order,

3

dq \dy J dp

+ rd/d/

dp dq

126 PAETIAL DIFFERENTIAL EQUATIONS [CH. XXVIII.

(fdF\ dF (&F\ dF}

\\--j-] -j- + \-I-\-T-\ =o,

\\dxj dp \dy J dq)

in which

fdF\ dF dF fdF\ dF dF

-j- ) = -r- + P~r> -J~ ) = T" + 2 T- •
\cte/ ax L dz \dy J dy * dz ,

Regarding the function F in the proposed integral F= 0
simply as a function of x, y, z, p, q, we have

fdF\ dFr dFg = Q

\dx j dp dq

J- (11).

(dF\ dF dF '

i — j — ) -j — j s ^ — j— t — u
\dy j dp dq

On the other hand, regarding F as a function of x, y, z, p, q,
mediately through u and v, we have the system

dF (fdu\ du du | d]? ((dv\ dv dv
du \\dxj dp dq } dv \\dxj dp dq

dF (fdu\ du du .} dF (fdv\ dv dv '

-T-^r j- ^r(j- ^r ^r
du \\dyj dp dq j dv [\dyj dp dq

and these systems are equivalent.

dF
Now if from the second of these systems we eliminate -j-

and -7—, we obtain (Art. 2), a result which must be equiva-
lent to the proposed partial differential equation,

(13).

This equation then considered as a relation between r, s, t,
must be an algebraical consequence of the relations (12), and

ART. 4.] OF THE SECOND ORDER. 127

therefore of the equations (11). If then we determine alge-
braically two of the quantities r, s, t, (we select r, t} from the
system, and substitute their values in (13), that equation
ought to be satisfied independently of the value of the re-
maining quantity s. Now supposing p and q to be both con-

. . dF dF

tained in F, so that neither -y- nor ,— vanish, we have

dp dq

from (11),

dF\ dF

fdf\ dt_

dF
dp

tdF\ dF
\dy J dp ""

dq

substituting which in (13) there results

fdF\ dF (dF\ dF (dF\ (dF\ dF dF
K L* J ^V~

{?? (dJL
} \dq

_ R ^ — T ( d— Y
dq dp \dp /

Now as this equation is to be satisfied in virtue of the con-
stitution of By S, T, U, V, and the function F, and indepen-
dently of s, both the coefficient of s and the absolute term not
containing s must be separately equated to 0. Thus F con-
sidered as a function of x, y, z, p, q, and containing p, q, at
least must satisfy the partial differential equations

fdF\dF (dF\dF

•" ~r~ ~i — \r J- [-y- \ ~r

\dx J dq \dy J dp

dp

dF dF

dF\dF fd_F\dF\
+~

(15).

128 PARTIAL DIFFERENTIAL EQUATIONS [OH. XXVIII.

This result may also be established by forming the equa-
tions of condition which express the proportionality of
It, S, ...V, to the corresponding quantities in the constructed
equation (7). From these equations of condition it is actually
possible to eliminate in two distinct ways the quantities

(dv\ (dv\ dv dv , , , . , f . c

I -3- i [~r-]t -7-5 -j- •> the result being the formation ot two
\dx) \dyj dp dq

partial differential equations for u agreeing in form with those
above given for F. (See the memoir Ueber die partielle Diffe-
rentialgleichung . , . Crelle's Journal, Vol. 61.) The actual
transition from the former to the latter rests upon the con-
sideration that the equation F (u, v) = 0, when F is arbitrary,
is not really less general than the form <1> {F (u, v), v} = 0, in
which the <5> is arbitrary. And here u has been replaced by
F(u,v).

The only condition respecting the application of the above
equations is that we do not admit any relations which make

... dF dF .

either -7— or -7- to vanish.
dp dq

5. PROP. III. The solution of the system of partial diffe-
rential equations established in the last proposition may in all
cases be made to depend upon that of simultaneous linear partial
differential equations of the first order.

In demonstrating this proposition we shall consider first
the case in which JJ— 0, then the case in which V= 0, lastly
the case in which neither of these quantities vanishes. The
ground of this division will appear in the investigation.

Case 1. Suppose U= 0. The equation then is of Monge's
form,

Hr+Ss+Tt= V.

The second equation of the system (15) becomes

ART. 5.] OF THE SECOND ORDER. 129

and therefore breaks up into the equations
dF dF dF dF

~1 mi -J- = °> -J m-2 J ~ = °>

dy 1dp dq 2dp

m1 and m2 being the roots of the quadratic equation

Bm*-Sm+T=0 (16).

As each of the above constituent equations is of the form

dF dF
-j- =m ~j— ,
dq dp

the system (15) may be reduced to the form

Em (—} — T (—} ~ -u Ym — — = 0
\dx J dp \dy J dp dp dp

which breaks up into the equations

j — vj •*-* *** I _7 I ' l 7 1 1^ " * 7 — *

dp \ax J \at/ J dp

The former of these we must reject (Art. 4). There re-
mains for the determination of F the system of linear partial
differential equations

**^m££m'i]

da dp

j- (17),

- Vm -j- = 0 I
dp J

and there will exist either one or two systems included under
this form, according as the roots of the quadratic (16) are
equal or unequal.

B. D. E. ii. 9

130 PARTIAL DIFFERENTIAL EQUATIONS [CH. XXVIII.

Case n. Let F = 0. The system (15) then becomes

fdF\ dF (dF\ dF fdF\ (dF\ _
'} L ^ U -

dp
TT((dF\dF fdF\ dF} .

+ u\ w-hr- + ( T- -j-t = Q'

[\dxjdp \dy J dq )

Eliminate Vby multiplying the first equation by

fdF\ dF^ fdF\ dF
\dx) dp \dy J dq '

the second by

fdj\ (dF\
\dx)\dy)>

and subtracting; we obtain, after rejection of the common

, dF dF
factor -j- -j- ,
dp dq

dF

We shall put this equation in the place of the second equa-
tion of the system. This we are permitted to do under the
restriction that in seeking to satisfy the system so changed
we do not make use of any relations which would cause either
of the two factors employed in the process of elimination to
vanish or become infinite.

The new equation reduces to one equation, or breaks up
into two equations of the form

©-©-

m being determined by the quadratic equation

Sm+ T=V.

A1IT. 5.] OF THE SECOND OEDEE. 131

/ '/•] 77\ //7 7it\

in (-— = m--

Making -^— = m-j-j ^ tne fi*8* equation of the system
(15), we get

dF\( dFdF dF\

which breaks up into

But if we combine the first of these with (18), we obtain

(D=°> ©-*

and this combination causing both the factors employed in
the elimination of U to vanish must be rejected. There
remains then the combination

(19),

and this will represent either one or two systems of equations
according as the quadratic determining m has equal or un-
equal roots.

Case m. Let neither £7=0 nor F=0.

Multiply the second equation of the system (15) by an
indeterminate quantity I, and add to the first ; then we have

J dp \dy J dp
mfdF\dF fdF\dF

Ul [ -j- I -j — \- M [-J- } -j-
\dyJ dq \dxj d%

9—2

132 PARTIAL DIFFERENTIAL EQUATIONS [CH. XXY1II.

We shall enquire whether it is possible so to determine I
as to resolve this into linear factors.

We might investigate this by resolving the equation as
a quadratic with respect to -j- or -y-. But the form of
the equation suggests what the forms of the linear factors
must be if the resolution be possible. For as the squares of -3—

and -j- both appear, and these squares alone, in the func-
tion to be resolved, it is clear that -=- and •T- will be the

dq dp

only differential coefficients of F which will appear in both
linear factors in common. The most general supposition

TTfl JTjl

possible is then that one factor shall contain -7- and ^- with

dq dp

- | , the other the same with ( -3- ) .

far \»//

Assuming then one factor to be of the form

,dF dF fdF\

I -j- + m -j- + n \-j~ I ,

dq dp \dxj

it is seen from the. form of the coefficients of the first three
terms of (20) that the other factor must be of the form

„ dF Tl dF U fdF\

J£ I L / J

dq in dp n \dy J '
and the resolved form of (20) must be

U^? W^.V/<ffi\UldF+mW+n/d^ =()

{ dq m dp n\dy}}\ dq dp \dx))

Multiplying out and equating coefficients, we obtain the
conditions

ART. 5.]

OF THE SECOND ORDEE.
Tin

n

ui-ul
IT'

133

TP

m

The third and fourth of these conditions are equivalent,
and give n = 1. The first and second are also equivalent,

T
and give m = y%. These values reduce the last equation of

condition to

so that I is determined by a quadratic. The resolved form
of equation (20) now becomes

1 7? dF± TTl dF^ TT (dF\\ \1 dF^ T dF^ (dF\\
Iti j- + ill j — F- U f-y- }[ \l -j- + -Tf -}-+ (-j- lr = $•

[ «2 "p \dy/) { d% U dp \d

To these results Tve may give a somewhat simpler form by
making Ul=m; not the m used above. We have then as
the quadratic for determining m,

- £77=0

(21),

and as the resolved form of (20),

&* m*£±nl**\\l dF.i.TdF+TT(d*'}\-0
-J-T + in -j— +1/1-5- r \ m T~ + •• j r ^ V T7~ / r "•

^2. ^P >^ ^J 1 d<l dP

134 PARTIAL DIFFERENTIAL EQUATIONS [CH. XXVIII.

Let ml and m2 be the values of m. Then we have from the
last the two distinct equations

\B**+& + u(d/}\L'lf'+T^ + U(d/}\ = 0,

{ aq 1 dp \dy J } [ 1aq dp \dxj)

dF Tj(dF\\ ( dF_ Td,F n (dF\\_
dp \dy J ) \ 2 dg dp \dx )}

and it is evident that these will be together equivalent to the
equations (15) from which they were derived.

Now to satisfy these equations simultaneously it is neces-
sary that we should equate to 0 one linear factor from each of
their first members. If we equate to 0 the first linear factors,
we have

dF dF fdF\
d([ 1 dp \dy )

dq 2 dp \dy J

whence, by subtraction,

.dF

This combination must therefore be rejected (Art. 4). For
the same reason must the combination formed by equating to
0 the second linear factors in the left-hand members of the
above two equations be rejected. There remains then only
the combinations formed by equating to 0 the first factor of
one of these members, and the second of the other.

Thus we should have the combination
TdF dF TT(dF\

<fy m*3p ^ (22^

dF ^dF ^(di

ART. 7.] OF THE SECOND ORDER. 135

with the combination which would be obtained from this by
interchanging m^ and mt.

6. It results from the foregoing investigations that the
function F is in all cases to be determined by the solution of
two simultaneous linear partial differential equations with
five independent variables. Xow the theory developed in
Chapter xxv. shews that the number of integrals of such a
system cannot exceed three. That theory enables us both to
determine what the number of integrals is. and to construct
the system of ordinary differential equations, reducible to the
exact form, upon which their discovery depends.

We have seen that the knowledge of two integrals u = a,
v = b of the system enables us to construct a general first
integral

of the partial differential equation (3). And the solution of
this first integral would lead us to the second integral which
is the final object sought. But the direct solving of a partial
differential equation of the first order which is not linear and
which involves in its actual expression an arbitrary function
is difficult, and happily it may be avoided here. The fol-
lowing propositions will enable us to accomplish the virtual
solution by a different solution, founded however upon the
same general principles.

7. PROP. iv. The integrals of the respective systems of
simultaneous linear partial differential equations upon which
the determination of F depends are so related that if from tiro
such respective integrals the values of p and g_ are determined,
they will render the equation

dz =pdx + qdy

integrable. And in the particular case in which the tico systems
become identical, any two integrals of the system stand in the
same relation.

136 PARTI AL- DIFFERENTIAL EQUATIONS [CH. XXVIII.

For, let <I> be an integral of the system (22), and "SP an
integral of the associated system obtained by interchanging
ml and 7W2 in the case in which these quantities are different.
Then <I> satisfies the equations

i

2 ay dp

and W satisfies the equations

' dV , dV TTfdV\
R -j- + mz -r- + U ,- = 0,
dq * dp \dy J

^-j- -j-
1 dq^ dp

But the necessary and sufficient condition in order that the
values, of p and q derived from the equations <E> = 0, M/1 = 0,
may render dz —pdx — qdy integrable, is

dx J dp dp \dx .

r Jfa //7\I/\

= 0 (23).

dy.
See Chap. xiv. Art. 11, Equation (36).

Now if from the previous equations we determine the values
of

V7 J J 1 7 I j 1 ~~? I * I -T" I «
dx) \dy J \dxj \dy J

and substitute them in the above equation of condition it will
be identically satisfied.

ART. 7.] OF THE SECOND ORDER. 137

The determination of [-^-] .... from the previous systems
\dxj

requires that U should not vanish. Hence the proposition is
established except in the case of £7=0, which is left doubtful.

To examine this case let us revert to the system (17) which
is proper to it. To that system since

whence Smjn^ = Ty

we may give the form

dF_ dF_ fdF\ fdF\ V dF_

dq l dp \dx J * \dy J R dp

or the form obtained from this by interchanging m1 and ma.

Substituting in these respective forms <I> and ^ in succes-
sion for F, we find

d$> d&

_ V dV
\dx ~ > \dy B dp '

and these values substituted in (23) reduce it to an identity.
Thus the proposition is established generally.

Lastly, as in the case in which the two roots of the quad-
ratic for determining m are equal, the two systems of partial
differential equations for determining 4> and ^ become one, it
follows that if from two integrals of that one system we can
deduce values of p and y these values will render the equation,

dz — pdx — qdy = 0
integrable.

138 PARTIAL DIFFERENTIAL EQUATIONS [CH. XXYI1I.

8. PKOP. V. When the system of simultaneous linear par-
tial differential equations determining F admits of two integrals
u = a, v = b, it will admit or will not admit of a third inte-
gral w = c, according as the roots of the quadratic determininy
m are equal or unequal.

The system in question, (22), becomes when we divide by

rr J -x t fdF\ , fdF\ A, . , „

U and write for — =— and -,— their full expressions
\dy ) \dxj

dF dF m^ dF It dF _
dy+% dz+ U dp* U~dq~

dF dF TdF ™*dF_
dx+P dz + U dp+ U dq~ '

or At^=0, A2F=0,

in which

d m d R d

d d T d mz d
+ Udp+ U dq

Hence the equation

(^
becomes

m —m dF f T

//€-, *'*o *«* / * JL

In this expression the coefficient of the first term only has
been calculated.

Now, by the theory developed in Chap. xxv. in order that
the two simultaneous partial differential equations should
have their full complement of integrals (three) it is necessary
that the above equation should be satisfied identically. This
involves three conditions, namely,

\RT. 8.1 OF THE SECOND ORDER. 129

the first of which is the one affirmed in the Proposition to be

necessary.

Secondly, it is to be shewn that if this condition be satis-
fied and it the system of given linear equations admit of two
integrals u — a, v = b, it will admit of a third.

Keplacing ml and m3 by m the system becomes
dF dF m dF £ dF_
dy + 2 dz + U dp + U dq ~

dF dF T dF mdF_
dx+Pfc + U dp + U dq =

Now if we construct from this the corresponding system of
ordinary differential equations, we shall find it to be

dz —pdx — qdy = 0,
dp- ^dx-^dy^O,

dq - ^jdx - jjdy = 0.

Xow it is impossible that the first of these equations should
be integrated without a previous determination of p and q as
functions of x, y, z, seeing that dx, dy, dz are the three differ-
entials entering into that equation. Such determination can
only come from the integration of the second and third equa-
tions of the system. But if these equations can be integrated
in the forms u = a, v = b, then u and v being particular values
of F satisfying the partial differential equations, it follows
from the last Proposition that the values of p and q which
they will yield will make the first equation integrable.
Hence if the system admits of two integrals it will admit of
three ; as was to be shewn. On the basis of these Proposi-
tions the theory of the second integration rests.

140 PARTIAL DIFFERENTIAL EQUATIONS [CH. XXVIII.

Theory of the Second Integration.
9. First suppose the values of m unequal.

Then MI = al, vl = bl "being the two integrals (and we have
seen that there cannot be more than two) of one of the systems
of linear partial differential equations, and u2 = aa, vz = bz those
of the other, the general first integrals of the given system
will be

The values of p and q determined from these will by
Proposition iv. render

dz —pdx — c[dy = 0

integrable, and the integral of this will be the general integral
of the proposed partial differential equation. For it will in-
volve explicitly or implicitly two arbitrary functions derived
from those in the first integrals.

It suffices however, following herein Charpit's method, to
combine one general first integral derived from the one sys-
tem with a particular first integral derived from the other
system, e.g. the integrals

The values of p and q hence derived, and employed as
before, will lead to a second integral involving one arbitrary
function and containing two arbitrary constants. This con-
stitutes a complete primitive from which the general solution
will be obtained by converting one of the arbitrary constants
into an arbitrary function of the other, and eliminating the
latter between the equation and the one derived from it by
differentiation with respect to that constant.

Secondly, suppose the values of m equal.

In this case we have but one system of partial differential
equations so constituted however that if it admits of two inte-
grals it will admit of three.

ART. 9.] OF THE SECOND ORDER. 141

Let u = a, v = b, w=c represent these integrals. Then if
from these we eliminate p and q we shall obtain a final inte-
gral of the form

.and this constitutes a complete primitive from which we shall
deduce the general integral by making b = <f> (a), c =i|r (a),
and eliminating a between the equations

at \x. y. a.

f\ J \ * •* '

0 =

da

To prove this let us combine the general and particular
first integrals

v = ^> (M), u — a.

The values of p and q hence obtained make

dz — pdx — qdy = 0

integrable, and the result can be no other than the remaining
integral w = c, or rather what this would become on eliminat-
ing^? and q from it. But since the equations by which this
integration are to be effected are equivalent to

u = a, v = <f> (a),

w will become a function of x, y, z, a and <f> (a). Also by
Charpit's method c is to be treated as a function of a, so that
ultimately we have the result above assigned.

have here supposed U not to vanish. If it do the
theory assumes another but simpler form. Let

be the two general first integrals. Then, since by the con-
dition at the close of Art. 2, if p be eliminated from these
equations q will also disappear, it suffices to eliminate them
together in order to obtain the general second integral.

142 PARTIAL DIFFERENTIAL EQUATIONS [CH. XXVIII.

10. Although the cases in which U= 0 and V= 0 have in
the foregoing sections been treated for simplicity apart, their
theory might have been deduced from that of the case in
which neither ?7rior V vanishes.

Thus to deduce the equations for the case of U= 0 elimi-

3TJ1 TTfl

nate from the general system (22) -y- and -j- in succes-
sion, and we find

.dF TT fdF\ rr*
(BT_ m^ _ _ ^ (_) +

(RT - »X) ~- + UT( ~ j - Um^ ( ~ } = 0.

But from (21) RT- m,mt = UV.
Substituting, and then dividing by ?7we find
~dF

>dF^
the equation determining ml5 mz being

This is equivalent to the results of Art. 5, Case I.

11. We found it necessary (Art. 3) in order that the gene-
ral partial differential equation of this Chapter should be satis-
fied by the envelope of a system of surfaces the equations of
which contain three parameters varying under two conditions
that the relation

should be satisfied.

It appears from Art. 8 that this is but one of three condi-
tions necessary and together sufficient for this purpose. The
formal conditions for every form of ultimate solution con-
sistent with the existence of a general first integral F (u, v) = 0
can be deduced in the same way.

CH. XXVIII.] OF THE SECOND ORDER. 143

[In the Bulletin de VAcademie Imperiale des Sciences de
St P/lersbourg, Vol. IV. 1862, there is an article entitled Con-
siderations sur la recherche des integrates premieres des equa-
tions differentielles partielles du second ordre, par G. Boldt
(Lu le 7 Juin 1861).

The article occupies pages 198 — 215 of the volume. Al-
though the name does not quite correspond, I consider that to
be a misprint, and I attribute the article to Professor Boole,
partly from the nature of the contents, and partly because it
is known by his friends that he was engaged at a time corre-
sponding to the date here given in the preparation of a mathe-
matical article in French.

The object of the article is to determine the conditions
necessary for the existence of a first integral of the equation

^ d*z ^ d*z ™ d'z

where R, 8, T, and TFare any functions of x, y, z, ~ and — ;

a x ay

and also to determine the conditions which must hold in order
that Ampere's method of integration may be employed.

In Crelle's Journal, Vol. LXI. there is an article by Pro-
fessor Boole, entitled Ueber die partielle Differentialgleichung
zweiter Ordnung Rr + Ss + Tt + U(s* - rf) = V.

The article is dated 1862 ; it occupies pages 309 — 333 of the
volume.

Among Professor Boole's manuscripts I found a memoir
very closely resembling the article in Crelle's Journal; it

144 PARTIAL DIFFERENTIAL EQUATIONS &C. [dl. XXVIII.

would appear that the memoir was drawn up with a view to
publication in the Transactions of some English Scientific
Society, and that this design was afterwards abandoned in
favour of the article in Crelle's Journal.

After some hesitation I have resolved to print this memoir.
Even if the memoir had been identical with the article in
Crelle's Journal it would have been convenient to the English
reader to be able to avail himself of the investigations ; and
the memoir contains remarks which do not occur in the article,
and which are interesting in connexion with the history of the
subject. There is some repetition of matter which has already
been given in Chapter xxvui.; but I was unwilling to impair
the completeness of the memoir by abridgment or omission.
Accordingly the memoir forms the next Chapter of the present
volume.

In Article 2 of the next Chapter will be found the pro-
cess to which there is an allusion towards the end of Article 4
of Chapter xxvui.

It is obvious that the subject of partial differential equa-
tions of the second order was much studied by Professor
Boole. The chronological order of his writings on the sub-
ject appears to be as follows :

1. Chapter XV. of the first edition of his work.

2. The article in the Bulletin of St Petersburg.

3. The memoir which forms Chapter xxix. of the pre-
sent volume.

4. The article in Crelle's Journal.

5. The Chapter xxvui. of the present volume.]

CHAPTER XXIX.

ON THE SOLUTION OF THE PARTIAL DIFFERENTIAL EQUATION
Er + Ss + Tt + U(s* - rt) = V, IN WHICH It, S, T, U, V
ARE GIVEN FUNCTIONS OF X, y, Z, p, q.

1. THE equation, the theory of the solution of which
I propose to consider in this paper, is remarkable from its
connexion with Geometry. If the equation of a surface
contain three constants which vary as parameters in sub-
jection to any two conditions connecting them, the gene-
rated envelope will satisfy a partial differential equation of
the above form. In other words any envelope of the surface

F(x, y, z, a, b, c)=0

formed by the variation of a, b, c in subjection to two con-
necting conditions

</>x (a, b, c) = 0, fa (a, ft, c) = 0

is necessarily an integral of a partial differential equation of
the form given above.

Now this theorem is the more important, because it is
only when three parameters in the equation of a surface
vary in subjection to two relations that the envelope pos-
sesses, irrespectively of the form of the connecting relations,
definite character. If there be but one connecting rela-
tion it is possible to determine that relation so as to make
ic envelope assume the form of any surface whatever, and
therefore the possible system of envelopes is in such case

B.D.E. II. 10

146 PARTIAL DIFFERENTIAL EQUATIONS [CH. XXIX.

unlimited. If there be three connecting relations the para-
meters become absolutely constant and no envelope exists.

The partial differential equation

U(s2-rt] = V

is remarkable also as including all the cases in which a
partial differential equation of the second order admits a
first integral of the form

u=f(v),

u and v being definite functions of x, y, z, p, q, and f (v)
arbitrary in form.

Neither of these statements is sufficiently general to con-
stitute a theory of the genesis of the partial differential equa-
tion under consideration, but the second one is more general
than the first, and is indeed sufficiently so to serve as the
ground of an investigation which connects the solution of
the equation in all cases with the satisfaction of a system
of simultaneous ordinary differential equations of the first
order and degree. And this is the ground upon which the
method of the paper will rest. I propose to shew, 1st that
the solution of the given equation on the assumption that
a first integral of the form u =f (v} exists requires the satis-
faction of a system of two partial differential equations of
the first order and second degree ; 2ndly that this system may
be resolved into four systems, each consisting of two partial
differential equations of the first order and first degree, two
of which systems are irrelevant and the other two relevant ;
Srdly that the solution of the two relevant systems ulti-
mately depends on the solution of a system of ordinary
differential equations of the first order, and that from these
ordinary differential equations the given equation of the
second order may be deduced independently of the assump-
tion above mentioned. 1 shall also discuss the theory of the
second integration. And I shall exemplify another method
of solution connected by a remarkable law of reciprocity with
the above method.

ART. 2.]

OF THE SECOND OEDER.

147

First Investigation.

2. PROP. I. lfu=f(v) be a first integral of the, equation
Rr+Ss+Tt+U(st-rt)=V ............ (1),

then will u and v, considered as functions of x, y, z,pt q, each
satisfy two partial differential equations of the form

/du\* „ (du\ fdu\ T(du\*

j-j + o^-l-i-l+^-j-l

\dx/ \dxj \dyj \dyj

—\ —

dxj dq

du fdu\ du fdu
dp(dx)+dq(dy

—

dy) dp

_

dy) dp dq

du\ , f du

du

du du

, . , fdu\ j (du\ j , du
in w fiich -=- and [-5- stand for -j-
\dxj \dyj J dx

respectively.

To demonstrate this proposition we shall form directly the
partial differential equation of the second order of which
u=f(v) is an integral* and, comparing that equation with (1),
deduce the conditions for the determination of u and v.

Differentiating u=f(v), first with respect to x and secondly
with respect to y, we have

du du dz du dp du dq
dx dz dx dp dx dq dx

^ i ^L ^_ i ^£ dp dv dq}
fa + ~fa dx + fy dx + dq dx)'

du du dz du dp du dq
dy dz dy dp dy dq dy

-, . . (dv dv dz dv dp dv dq}

— — + [ j» ] j _j . i _r i . £L

7 W [dy dz dy^dpdy- dq dy]'

10—2

148 PARTIAL DIFFERENTIAL EQUATIONS [CH. XXIX.

.,, . du du , fdu\ du du . fdu\

or, if we represent -j- + p -j- by -j- , -7- + a -r- by ( -7- ,
dx ^ dz \dx) dy * dz J \dy]

•£- by r. -j- and §± by s, and -=± by £,
dx J dy dx dy

du\ du du „, . . f/c?y\

rT" +S7T=/ (v) 1 hr
<3^> ag Ivw?/

du du ,,, . {fdv\ dv dv\

s j- + t -7T=J (v) 1 T- l + s T- + ^ j-f •
d!p rt^ y \\dyJ dp dq)

Eliminating /'(v) we arrive at the partial differential equa-
tion of the second order,

(du fdv\ dv /du\~)

J _ I _ 1 ___ . __ , / -- I > 7*

(dp \dyJ dp \dyJ )

(du (dv\ dv /du\ dv fdu\ du, (dv\\
(dq \dy) dq \dy) dp \dx) dp \dx))

(dv /du\ du fdv\\ (du dv du dv} . 2 ^
\dq \dxJ dq \dxJ ) \dq dp dp dq}

du\ fdv\ fdu\ f
dy) (dx) ~ (dx) (dy

It is seen that as respects the mode in which the quan-
tities r, s, t are involved this equation is of the same form
as the given equation (1). That it may be equivalent, its
coefficients must stand to those of (1) in a common ratio /*.
This gives

du fdv\ dv fdu\

-7- -H —-T- -7- ==/*./£ ................................. (a),

dp \dyj dp \dyj

du /dv\ dv fdu\ d^fdu\ du fdv\ _ „ ,,.
dq (djj) ~dq (fy) + dp'(fa)~dp(d^)=IJ'L

dv /du\ du /dv\
dq (dx) ~Tq (dx) =

ART. 2.] OF THE SECOND ORDER. 149

du, dv du dv

,
du\ /dv\ /du\ /dv\ T7

%) (a) - (i) (#; ^ .......................... w-

As we have here five equations "which are homogeneous with
respect to the four differential coefficients of v and to #, it is
clear that we can, by the elimination of these quantities,
obtain a relation connecting the differential coefficients of u
with R, S, T, &c. But the peculiar cyclical form of the
functions in the first members of the above system enables
us to effect this elimination so as to lead to two final equa-
tions independent of v and p.

mi I.- i • / \ v fdu\ du . fdu\ du

Thus multlplymg (a) by , (c) by -. ,

du du
) ^ Tp ~dq_
rejecting the common factor /*,

\ du rrfdu\ du TTfdu\ (du\

-j- I 1 -- r •*• I ~T~ \ ~i -- r ^ \ ~T~ I \ ~7~ I

dxj dq \dyj dp \dxj \dyj

vdu du

+ r-j- -r = v
dp d'i

Again, multiplying (a) by , (i) by , (c) by

i / x i , ,.. ,

^J ' and w b^ v^J ^ + v^; ^ ' addm^ and asam re-

jecting the common factor /A, we have

dy) \dy

PARTIAL DIFFERENTIAL EQUATIONS [CH. XXIX.

Hence, u considered as a function of x, y, z, p, q satisfies
the two partial differential equations (4), (5), both which are
of the first order and second degree.

As u and v enter symmetrically into the system (a), (&),
&c., v will also satisfy two partial differential equations of
the same form, viz. the equations

fdv\ dv 7, AM dv
\dx) dq \dy) dp

dxj

'dv\ idv
dxj \dy

dv
dx) \dy

ydv_ dv^ ^^
dp dq

~r
dy

.

... (6).

dq

Further, these two systems of equations constitute the
complete system of equations resulting from the elimination of
/n from the five equations (a), (V), (c), &c. ; for in their deter-
mination, no factor involving either the differential coefficients
of u and v, or the quantities R, S, T, &c. has been rejected
directly or indirectly.

I am not aware that the above results of elimination have
been noticed before.

3. PROP. II. The system of partial differential equations
above obtained for the determination ofu, viz.

du\ du

j-J j~

ax/ dq

dx

du\ du
dy) dp

Tj fdu\ /du\
\dx) \dyj

du du
-r-T =0
dp dq

du\ (du\

-3~}ly)

dxj \dy/

T(du\9
\dy)

....(7)

ART. 3.] OF THE SECOND ORDER. 151

admits of resolution, into four systems, each consisting of two
linear partial differential equations ofthejirst order. Of these
systems two only are relevant to the solution of the problem.

For, multiplying the second by an indeterminate quantity
X, and adding the result to the first, we have

i\ du

dp dq T """ \dy) dp

j dp \dyj

Xow let us see if it is possible to determine X so as to
make the first member of the equation resolvable into linear
factors. We cannot say a priori that such resolution is pos-
sible as we should be able to do if that member were homo-
geneous and of the second degree with respect to three instead
of with respect to the four subject variables

/du\ fdu\ fdu\ fdu
(dx)' (dy)' \dp)' (dq

du\

Observing that the squares of -y- and y- are wanting

in the first member of (8) while those of f-j-J and [-j-j

\dxj \dyj

appear, we are led to assume as the proposed equivalent of
that member an expression of the form

Multiplying the factors of this expression together and
then equating the coefficients with those of the first member
of (8) we have

152 PARTIAL DIFFERENTIAL EQUATIONS [CH. XXIX.

Rm' + \m = 17+ S\ ..................... (a),

mm'=T\ .......................... (&),

\n = \V=mn .......................... (c),

Rri = R ............................. (J),

nm'=T ............................ (e),

nn'=V ........................... (/),

From (J), (c), (d], we find

T

n— V, n = 1, m = \V, m = — ,

values which will be found to satisfy (e) and (/) also, and
which reduce (a) to the form

V*\*-SV\ + RT- £77=0.
Supposing A, thus determined, the equation (8) becomes

( - fdu\

\R(-r)
I V«ay

du\ ~fTdu} ( fdu\ T fdu

+ \-r- } + Vj- X -- -

»?// "

The result is a little simplified if we retain m in place of X.
We thus find as the resolved form of the given equation

T

m being determined by the quadratic
m*-Sm+RT-U

If mt , m2 be the values of m thus found, we have

du\ T7du\ ( fda du

7 - + V-j-\ \mt [-T-
«y/ cjpj I \»«

du ,du ( dn

-j = 0,

\dijj dq)

ART. 3.] OF THE SECOND ORDEK. 153

and these two equations are manifestly together equal to the
system (7).

Xow these equations can only be simultaneously satisfied
by equating to 0, one factor in the first member of each ; and
the different combinations which are thus possible give rise
to four binary systems of linear equations. Let us examine
these systems separately.

If we simultaneously equate to 0 the two first factors of
the left-hand members of the last two equations, we have the
systems

fdu\ fdu\ rrdu

( T + mi (-J- + F:r = °>

\dxj l \dy) dp

a system which, when m^ and mz are different, is reducible to
the system

D fdu\ r,du fdu\

R( -r)+ F-T- = 0, (— =0.
\dx/ dp \dy/

It is clear that this cannot lead to a value of u satisfying
the given differential equation (1), because it takes no
account of the forms of S, U, and T. Indeed if we actually
eliminate

du\ /du\ du du
}' \fy)> ~fy' dq

from the above equations by means of the system

/*,\ 4, *,

J1 ................ (10),

d>i\ du du

--r
di// dp

154 PAKTIAL DIFFERENTIAL EQUATIONS [CH. XXIX.

(derived from the assumed first integral u =f(v) by making
f(v) = c, and differentiating the result first with respect to x,
then with respect to ?/), we find as the result

Vt + JR (s2 - rt] = 0.

Again, if we equate to 0 the two last factors of the right-
hand members of (10), we have

m (^]+T(-
1 \dx/ \dy,

du\ m fdu\

fdu\

\-j- }
\dxJ

--

\ 7 / I ' 1 J

\dy/ dq
which, if 7WX and mz are different, reduce to

And it is evident that neither are these equations consistent
with the given equation (1), because they take no account of
S, U, and R. The equation of the second degree to which
they actually lead is

There remain then the two systems formed by combining
the first factor of each one of the first members with the
second factor of the other, viz.

.„ fdu\ fdu\ r7du

E [ -=- + ™i ( T- ) + ^ J- = °

\dxj \dy] dp ,

\dx) \dy) dq

vfdih . fdii

2t( -7-

Y -, ' =

I 7^ I T^ v/t, l -j I T • 7

\axj \di/J dp

}• (12).

du\ T(du\ ydu_

\dy) dq

ART. 3.] OP THE SECOND ORDER. 155

That these systems are relevant to the solution of the pro-
blem under consideration may be shewn by eliminating from
either of them by means of (10) the quantities

fdu\ fdu\ du du

UK/' \dyr Ap* ~dq'
The actual result will be

V{Br + Ss+Tt + U(s* - rt) -V} = 0 .......... (13),

which, except in the particular case of F= 0, reduces to the
given equation.

More generally, if in the equation

"=/(')

u and v are any distinct solutions of the system (11), the
same result of elimination may be deduced. For v by hypo-
thesis satisfies the equations

Subtract these equations multiplied byf'(v) from the corre-
sponding equations of (11), and representing u—f(v) by W,
we have

which being of the same form as (11) it follows that
TF=0 or u-f(v) = 0

also leads to the partial differential equation of the second
order (13).

156 PARTIAL DIFFERENTIAL EQUATIONS [CH. XXIX.

4. PROP. ill. To reduce the determination of the first inte-
grals of (1) to the solution of a system of ordinary differential
equations.

Each of the systems (11), (12) presents u as satisfying
simultaneously two linear partial differential equations of the
first order.

To deduce the value of u thus conditioned it will obviously
suffice to multiply in each system one of the partial differen-
tial equations by an indeterminate multiplier X, to add the
result to the other equation so as to form a new equation
which will, like those from which it is formed, be linear and
of the first order, and which on account of the indeterminate
character of X will be equivalent to the two. From the
auxiliary equations which we obtain in the process of solu-
tion, X must be eliminated.

If in this way we combine the equations of the system (11),
we have, on arranging the resulting equation according to the
differential coefficients of u,

p + mA + X ( Tq +m,p)} ~

rrdu ^ Trdu
+ VT- + \Vj- =
dp ay

Hence we have the auxiliary equations

dx dy _ dp _ dq __ dz

1R + Xw ~ m + \T~ ~V ~ ~

and it is to be remembered that w,, w2 are the roots of the
equation

ART. 5.] OF THE SECOND ORDER. 157

Eliminating X from the first four of the above equations we
have

Udq + m^dx - Edy = 0]

Udp + mtdy - Tdx = 0\ (I).

dz — pdx — qdy = OJ

This then is the system of ordinary differential equations
deduced from (11) upon the integration of which the determi-
nation of u will depend.

A similar system, differing from the above only in the
mutual transposition of ml and m2, is given by (12), viz.

Udq + madx - Edy = 0 j

Udp + m^dy — Tdx — 0? • (H)-

dz — pdx — qdy = 0)

If from either of these systems we can deduce two inte-
grals of the forms

u = a, v = o,

it is obvious, from what precedes, that

u =f(v)

will constitute a first integral of the proposed (1), and there
being two systems in question, two such first integrals, each
involving an arbitrary constant may coexist.

5. PROP. IV. To deduce the second integral of (1).

It will be necessary to consider separately the cases in
which m1 and ma are equal and unequal.

First let m1 and m.^ be equal.

Both the systems (I), (II) reduce to a single system which
may be expressed in the form

21, m ,

Bdy r (»)•

dz = pdx + qdy

158 PARTIAL DIFFERENTIAL EQUATIONS [CH. XXIX.

Now. since the condition ~ = -~ is here satisfied, it is

ay ax

manifest that if from any two integrals of the above system
of the forms u = a, v = b, simultaneous values of p and q 'be
determined, these values will render the third equation of the
system integrable, and the effect of its integration will be
virtually to determine z as a function of x, y, and three arbi-
trary constants, viz. a, b, and a constant c introduced in the
last integration. Let us represent the result in the form

z = (f>(x, y, a, b, c] ..................... (15).

Now what relation will this result bear to the general solu-
tion of the partial differential equation given, to the solution
which we should obtain by integrating, not the particular
equations u = a, v = b, but the general first integral u =f(v),
which includes them both.

To integrate the equation u =f(v] it suffices to deduce any
particular equation involving an arbitrary constant b, which,
in conjunction with u =f(v) will render

dz — pdx — qdy = 0

integrable, and to integrate the last equation regarding the
arbitrary constant of integration as an arbitrary function of
b. The result is a complete primitive in which, by the
variation of b as a parameter the general integral is implicitly
involved.

Now either of the equations u = a, v — b will, in conjunc-
tion with u =f(v) determine p and q so as to make

dz —pdx — qdy = 0
integrable. Take the equation v = b, then u =f(v) reduces to

Thus, in place of the equations u = a, v = b, of the previous
section, we have

ART. 5.] OF THE SECOND OEDER. 159

for the determination of p and q. The constant c introduced
in the final integration becomes also, according to the above
theory, a function of b, and the complete primitive is of the
form

while the general integral is found by eliminating b between
this equation and its differential with respect to b.

The general integral therefore represents the envelope of
the surface represented by (15), a, b, c being parameters sub-
ject to any two connecting conditions.

As ml , m, are supposed equal, a necessary condition of the
Ability of this species of integration is that

S*--±(IIT- CT) =0 ................... (17),

a
the value of m is — , and the system (14) reduces to

dz — pdx — qdy = 0

We conclude therefore that if (17) be satisfied and ice can
from (18) deduce a value of z in terms of x, y, and three
arbitrary constants, the equation expressing that value will be
a complete primitive, and the general integral will be found
by making the constants vary in subjection to two arbitrary
conditions.

Ex. Let the given equation be

xyr + ypt + xy (s~ - rt] =pq.
Here E = xq, S=Q, T=>yp. U=xy, V=pq.

160 PARTIAL DIFFERENTIAL EQUATIONS [dl. XXIX.

The condition (17) is satisfied, and (18) becomes
xydp —ypdx = 0,
xydq — xqdy — 0,
dz —pdx — qdy = 0.

•

From the two first of these we find
p = ax, q = ~by,
whence from the third,

axz bi/2

This is the complete primitive, and the general primitive
consists of all possible equations derived from this by making
a, b, c vary in subjection to two conditions.

Ex. 2. Given

Here the equation for m reduces to

w2 + "2pgm +p*(f = 0,
whence m = —pq, and the system (18) gives

— - r +P<ldy + (1 + p*) dx = 0.

ART. 0.] OF THE SECOND ORDER. 161

Subtracting the upper equation multiplied by pq from the
lower one multiplied by 1 + j2, and dividing by
•we have

whence

In like manner,

ft

—^ = 5.

+ 22)1

Hence determining p and £,
,

Therefore (a; - a)s +(y- &)2 + (a - c)1 = 1 .

From. this form of the complete primitive it is evident that
the general integral will represent all possible tubular surfaces
formed by the motion through space of a sphere of constant
radius unity.

Secondly, let ml and mz be unequal.

Then since, in neither of the systems (I) and (II) is the
condition -j- = -~ satisfied, from neither system separately

can values of p and q be obtained which make dz =pdx + qdy
integrable.

But, as will be shewn, any two integrals obtained, the
one from the one system and the other from the other, will
give values of p and q which will render dz = pdx + qdy in-

B.D.E. II. 11

162 PARTIAL DIFFERENTIAL EQUATIONS [CH. XXIX.

tegrable, and the general solution will consist of all possible
integrals of the latter equation thus obtained.

Or if the complete first integral of either system be com-
bined with any particular integral involving an arbitrary
constant obtained from the other, the two will furnish values
of p and q which render dz = pdx + qdy integrable, and its
integral will be a complete primitive involving one arbitrary
function in its expressed form, another in the connexion of
its two constants; the general primitive being found in the
usual way by making the constants vary as parameters in
subjection to a single arbitrary connecting condition.

In fact it may be shewn that if we attempt by the process
of Charpit or Lagrange to integrate the partial differential
equation of the first order u=f(v), deduced we will here sup-
pose from the system (i), we virtually construct the system
(11) in the auxiliary equations upon which the process of
solution turns. I have obtained a direct proof of this proposi-
tion, but I think it preferable and at the same time sufficient,
to direct attention to the prior ground upon which it rests in
the relations of the systems of partial differential equations
(11), (12) from which the systems of ordinary differential
equations (i), (n) are derived.

Let P=0 represent any integral of the system (11), and
Q = 0 any integral of the system (12). Then we have

dp \dxj 1 \dy)

dQ_ dQ

ART. 5.] OF THE SECOND OBDER. 163

Hence we deduce

v(dPfdQ\ dQ fdP\ JP(W\ _3Q
V \dp\dx) dp(dx) + dq(dy) dq

The second member of this equation is identically 0. Hence
dividing by V we have

_ _

dpdx dp\dx-dq\dy dq \dy

But this is the known condition tinder which the values
of p and q deduced from the equations P = 0, Q = 0 make
dz =pdx + qdy integrable; see Chap. XIY. Art. 13, Equation
(36).

TT> conclude then that if from the systems (i), (n) we can
deduce two corresponding systems of integrals

then will the first integrals of (1) be

while the second integral will consist of all possible relations ob-
ta ined either 1st by specifying the forms off^ , ft and obtaining p
and q as functions of ' x and y and integrating dz =pdjc + qdy,
or "Zndly, by specifying one of the functions f^, fs, leaving the
other arbitrary, determining p, q, integrating dz = pdx + qdy,
and regarding the final constant of integration as an arbitrary
parameter.

11—2

164 PARTIAL DIFFERENTIAL EQUATIONS [CH. XXIX.

Ex. Given ar + ls + ct + e (s2 - rt] - h, the coefficients
being constant.

Here£=a, 8 = 1, T=c, U=e, V=h.
Hence w», . mz are the roots of

w2 — 5m + ac — eh = 0,
and the systems (i), (n) give

edq + m^dx — ady = 0}

ay = 0}
— cdx — §\ '

edq + mzdx — ady = 0
edp + infy — cdx = O

Whence the first integrals are

«2 + m^x - ay =f1 (ep + m2y - ex],
ec[ + mjc - ay =f2 (ep + mjj — ex),

from which all possible second integrals are to be derived in
the modes above explained.

Let us take the second of those modes and give to the
second of the above first integrals the particular form

ep + tnjj — cx= C}

G being an arbitrary constant. From this, and from the
other integral, left in its complete form, we have

ex - my + O ay- mx+f {(m - m] y + C]

= — -- -—-

whence, substituting in the formula dz =pdx + qdy, inte-
grating, replacing the arbitrary form

ART. 6.] OP THE SECOND OEDER. 165

and introducing an arbitrary function of C for the arbitrary
constant, we have

complete primitive. The general primitive consists of
all possible relations obtained by eliminating C between the
above equation and

when the forms of <f> and ^ are specified.

Second Investigation.
6. If from the equation

rt) = V. (20),

we eliminate r and t by means of the equations
dp = rdx + sdy,
dq = sdx + tdy,
the result will be

[Rdif - Sdxdy + Tdx* - U (dpdx + dqdy)\ s

= Bdpdy + Tdqdx - Udfdq - Vdxdy (21).

There are different considerations (all of them however in-
volving, as I have been led to think, a more or less explicit
reference to some theory of the genesis of the given partial
differential equation) which indicate that its solution depends
upon that of the equations obtained by equating to 0 the part
affected and the part not affected by s, viz. upon the solution
of the equations

Rdy1 - Sdxdy + Tdx* - U(dpdx + dgdy] =0 (22),

Elply+ Tdidx- Udpd% - Vdxdy = 0 (23).

166 PARTIAL DIFFERENTIAL EQUATIONS [CH. XXIX.

Without entering into these considerations let us inquire
what consequences may be deduced from these equations
assumed to be true.

It is seen that these equations are connected by a remark-
able reciprocity with the partial differential equations (7).
They will in fact be converted into these equations if we
change

dx, dy, dp, dq, U, V, S t...- (24),

into

du du du du T/r TT ~ , ,

~%'~^' dq> dp> V) U>~

respectively. From this formal connexion it follows that if
we multiply (22) by X and add to the result (23), we shall be
able to determine \ so as to permit the resolution of the equa-
tion thus formed into linear factors. Ultimately we shall, as
appears from Art. 3, reduce the system (22), (23) to an equi-
valent system of the form

(- Rdy - m^dx + Udq) (- m^dy - Tdx + Udp} = 0,
(- Edy - mzdx + Udq} (- mzdy - Tdx + Udp} = 0,
m1 and mz being determined by the equation

or, changing the sign of m,

(- Edy + m^dx + Udq} (m^dy - Tdx + Udp} = 0
(- Edy + m2dx + Udq} (m2dy - Tdx + Udp] = 0

ml and m2 being as in the former investigation roots of

Equating to 0 the corresponding factors of the first mem-
bers we have

— Edy + m^dx + Udq = 0

— Edy + madx + Udq = 0

m^dy — Tdx + Udp = 0
m2dy — Tdx + Udp

\

= OJ

AKT. 6.] OF THE SECOND OKDEE. 167

The first of these, ml and ma being different, is resolvable
into

Udq - Edy = 0, dx = 0 ;

the second into

and it is obvious that neither of these can lead to the given
partial differential equation (1). The first of them combined
with the equations

(27),
leads in fact to the partial differential equation

E- Vt = 0 .............................. (28), .

the second in like manner leads to

T- Ur = 0 ........... . ................ (29).

But equating to 0 the non-corresponding factors of the first
members of (26) we have

— Edy + mfa + Vdq = 0
mady — Tdx + Udp

mfy-Tdx+Udp^O
- Edy + madx +

-01

=or

rdp = 0|

W<7 = O) '

Now these systems when completed by the equation
dz = pdx + qdy agree with the systems (i), (il) deduced in the
previous investigation.

It remains to shew that these systems actually lead to the
given partial differential equation (1) directly. Eliminating
from either of them, combined with the system (27) the dif-
ferentials dx, dy, dp, dq, we shall have as the result

U{Rr + Ss+Tt+U(s*-rt)-V} = 0 ......... (30),

which, rejecting the factor U, as from (13) we rejected F, is
the differential equation proposed.

168

PAETIAL DIFFEEENTIAL EQUATIONS [CH. XXIX.

Ground of the Reciprocity above noticed.

7. The reciprocity above noticed is not of a primary cha-
racter, but is founded upon two prior laws which I shall pro-
ceed to demonstrate.

If from the partial differential equations of the -system (7)
we eliminate Fand substitute the resulting equation in the
place of the first equation of the system we shall obtain the
equivalent system

'du

dq

du

,du du
dqr dp

T (du
1 (dp

du\ du

du

...(31).

These equations are both symmetrical and it will be ob-
served that they are convertible the one into the other by
changing

du du * du

into

dq ' dp' dx

du du du
dx' dy' dq'

5F, u, s, v

dy

du
dp'

V, -S, U

(32),

(33),

respectively. This is a law of reciprocity which connects
solely the differential coefficients of u and the coefficients
Z7, S, V of the original equation.

Again u = 0 is by hypothesis a solution of the given partial
differential equation. Regarding it however simply as an

ART. 7.] OF THE SECOND ORDER. 169

equation which is true and the truth of which is consistent
with that of the equations

dp = rdx + sdy)
dq = sdx + tdy)

and differentiating it first with respect to x, secondly with
respect to y, we have

du du dz du dp du dq _
dx dz dx dp dx dq dx '

du du dz^ du dp du dq _
dy dz dy dp dy dq dy

equations to which we may give the form

/du\ _ du du
\dx) dp dq

fdu\ _ du du
\dyj dp dq '

Now this system is of the same form as the system (27)
and will agree with it if we change

du du du du /0_N

— — — =- . -y- -y- (v<>.)>

dx dy dp dq

into

dp, dq, dx} dy (36),

respectively — a change which does not affect the coefficients
of the given equation, and which is therefore the expression
of a law of reciprocity distinct from that last noted. The
combination of these two laws does however lead to the
law exemplified in the researches of the previous Article ;
see (24) and (25).

The question here arises whether it would not have been
better to employ from the first the symmetrical forms (31) of

170 PARTIAL DIFFERENTIAL EQUATIONS [CH. XXIX.

the partial differential equations of the first order and second
degree upon which u depends, than the unsymmetrical forms
(7). It was indeed from the symmetrical forms that the
chief results of this paper were originally obtained, but the
unsymmetrical forms lead to the same end in a simpler way,
and therefore they have been made use of in the present
memoir.

It may be proper to notice, in concluding this section, that
the symmetrical forms in ordinary differentials would have
emerged in place of the unsymmetrical ones of (22) and (23),
if the quantity s2 — rt had been retained instead of s. The
equations

dp — rdx + sdy, dq = sdx + tdy,

enable us in fact to reduce the given equation (20) to the
form

Rdp* + Sdpdq + Tdq* - V(dpdx + dqdy}

= (s2 - rt} {Edf - Sdxdy + Tdx* - U (dpdx + dqdy}}.

Hence arises the symmetrical system

Rdp* + Sdpdq + Tdq* - V (dpdx + dqdy} = 0,
Rdy* - Sdxdy + Tdx* - U (dpdx + dqdy} = 0,

which is connected with the system (31) by the single law of
reciprocity expressed in (35) and (36).

Postscript.

8. At the time when the above investigations engaged my
attention I was totally unaware that the subject of them had
been discussed by Ampere (Journal de VEcole Poly technique,
Tom. XI.) and recently by Professor De Morgan ( Cambridge
Philosophical Transactions, Vol. IX. Pt. IV.). I feel it thnv-
ibre incumbent upon me to state why after acquainting my-

ART. 8.] OF THE SECOND ORDEB. 171

self with the results of their labours, I offer this paper for
publication.

The method of Professor de Morgan so far resembles the
first method of this paper, and that of Ampere the second,
that while the former makes the solution of the problem
depend directly upon that of simultaneous partial differential
equations of the first order, the latter makes it to depend
directly upon the solution of simultaneous ordinary differ-
ential equations of the first order. The formal connexion of
these methods by the law of reciprocity is, I believe, esta-
blished for the first time in this paper. The system of partial
differential equations of the second degree (7) has not, so far
as I am aware, been given before.

But a point which I think of deep importance is the follow-
ing. By connecting, as in this paper, the differential equa-
tions of the second degree, whether ordinary or partial, by an
indefinite multiplier which is afterwards determined so as to
admit of the resolution of the system into its component linear
elements, we assure ourselves that each step of the solution
offers a complete sequence to that which has gone before, and
it only remains then to separate the different elements and
determine whether they are relevant or irrelevant to the end
in view. That any such distinction exists has not, so far as
I am aware, been noticed before. And it seems to me the
more important that it should be noticed because the solution
of partial differential equations in cases far more general than
those above considered seems to depend upon the satisfaction
of simultaneous differential equations of a degree higher than
the first. I have in fact by an application of the Calculus of
Variations arrived at the conclusion that the theory of the
solution of all partial differential equations of the second
order, whatever the number of variables may be, is very inti-
mately connected with the satisfaction of a system of differ-
ential equations of the type

dF... dF , dF ,

F=Q representing the given partial differential equation, x
and y any two of the independent variables, and r, s, t the

172 PARTIAL DIFFERENTIAL EQUATIONS [CH. XXIX.

second differential coefficients of the dependent variable with
respect to x and y.

I may perhaps at some future day resume the subject, to-
gether with an inquiry into the theory of the solution of the
partial differential equation of this paper, when the conditions
under which the auxiliary equations (i), (n) are supposed to
be integrable are not satisfied.

9. NOTE. It may be desirable to establish directly the
converse form of one of the results of Proposition IV. For
this object we shall shew that the equation of the envelope of

z = (f>(x, y, a, b, c) ........................... (1),

where a, b, c are connected by any two conditions of the
forms

-^ («> b, c) = 0, x (a> b, c} = 0,
will satisfy a partial differential' equation of the form

Er+Ss+Tt+U(s*-rt)=V ............... (2),

in which also

Differentiating (1) we have

_ d(f> d$> da d(f) db d<f> dc

* dx da dx db dx dc dx

d(j> d<j> da d(j> db d$ dc

*> dy da dy db dy dc dy

and by the nature of an envelope these reduce to
dd) dd>

Again differentiating these equations with respect to x and
, and writing for simplicity

dadx ' dbdx '

^2<ft _ ^ <P$ _ £• dty_
dady dbdy dcdj

AET. 9.]

we have

OF THE SECOND ORDER.

dx

173

S =

dxdy dy dy dy '

d*$ ,da iS,db + c,dc
dxdy dx dx dx'

f-

t =

'da

Hence we find

s —

dxdyj

A _L -R

— La — h-o-

db
^-
CUE

, mn>

+ (XJO —

dy) \ dx
,da

dy

da db da db
dy dx dx dy.

C}(— — _-- — }
\dy dx dx dy)

da dc i

dx

^
~r
dy

T
dy

Now since a, b, c are connected by two conditions, so that
b and c are functions of x and y only as being functions of a,
we have

da db da db db dc db dc _

dy dx dx dy dy dx dx dy

dc da dc da _
dy dx dx dy

174 PARTIAL DIFFERENTIAL EQUATIONS. [CH. XXIX.

Thus the above equation reduces to

/ dW / d*d>\f d*d>\

(s-T-H ~ \r--~ri (<- T-« = °>
V dxdy) \ dx2 J\ dy*J

or

_

df dxdy dx* ~dtf ~ty

This equation is of the general form (2). Its coefficients

72 J

-y^- , &c. are determinable as functions of x, y, z, p, q when

the form of the complete primitive (1) is given. For this
purpose the complete primitive with the two derived equations
(3) suffice.

Again, comparing (4) with (2) we have as the conditions
of their equivalence

E s v

_

_
dy* dxdy dx2 dx* dy* \dxdy

conditions which suppose R, S, T, V, V connected by the
relation

( 175 )

CHAPTER XXX.

ADDITIONS TO CHAPTER XVII.

[THE present Chapter consists of additions to Chapter XYII.
Art. 1 was intended to follow Chap. xvn. Art. l.J

1. The theory of the solution of linear differential equa-
tions in a series flows very beautifully from their symbolical
expression. It is usual in treating this subject to assume the
form of the series, and deduce from the differential equation
the law of its coefficients ; but the symbolical form of the dif-
ferential equation determines in reality the form of the solu-
tion as well as the law of derivation of its successive terms.

Let us begin with the binomial equation
f0 (D) u -/ (D) e^u = 0.
Operating on both sides with {f^D}}'1, we have

in which

Hence (l-<^) e> = {/0(Z))P<).

Now {/oOD)}-1*) will be determined by the solution of a
linear differential equation with constant coefficients, and will
be necessarily of the form

AP+BQ+CB+...,

in which A, B, C, ... are arbitrary constants, and P, Q, .#, ...
are functions of the independent variable.

176 SYMBOLICAL METHODS. [CH. XXX.

We have then

(1 - <f>(D) ere} u =AP+SQ + CR+ ... ,
therefore u = {\-$(D) e'Y (AP+BQ + CR+ ...).

Now let us represent (f> (D) er0 by p ; then

Represent the first line of the above expression by u},
then since

pm = <j> (Z>) er9<j> (D} ere . . . m times

= 6mre<j>(D + mr)<f>(D + mr-r) ...... <f>(D + r),

we have

Ul=A{P+er6<l>(D + r)P+e*r<)<j>(D + 2r)<f>(D + r)P
+ <?re<j>(D + 3r}<j>(D + 2r}<f>(Z> + r) P+ ...},

in which it only remains to perform the operations indicated
by </> (D + r), by <£ (D + 2r) j> (D + r), . . . on the function P.

Let us in the first place suppose the symbolic function
/0 (D) to be of the form (D - a] (D - ft) . . . ; then

Here P= ea6. Hence substituting in the above expression
for u, and observing that /"(I)) e™6 =f(n) e"e, we find

w, =.
or, since e9 = x,

ART. 1.] ADDITIONS TO CHAPTEE XVII. 177

and

u =Ax* [I + </>(a + r) x'+ <j>(a + 2r) $(a + r)

the solution sought.

Consider now the general equation

Here we have, representing 7-/7JS t>7 <f>m(D),

therefore

Here we have first to determine {f^DYf1^, then to deter-
mine the effect of the operation represented by

upon this.

Xow [ft(D)}~*Q is given by the solution of a linear diffe-
rential equation with constant coefficients, and will therefore
be of the form

AP+BQ+CR+ ..., .

A, B, C, ... being arbitrary constants, and P, Q, E, ... func-
tions of 6.

Again, since

it may be shewn by a process of actual symbolical division,
B.D.E. II. 12

178 SYMBOLICAL METHODS. [CH. XXX.

attending to the laws of combination of symbols, that the
expression may be expanded in the form

To determine the functions F0(D), F^D), ...... we may

proceed as follows. From the equation

we have

l}}e' + ................ (1).

Hence F0(D) = 1,

therefore F, (D} = - & (D) F9 (D - 1) ,

and so on. Hence FQ(D}, F1(D], ...... are determined in suc-

cession. The general law is as follows : the coefficient of
eme in the second member of (1), when m is greater than 1, is

(D-2} + ... (2),
whence

By this formula the successive values of Fm(D) can be
deduced from those of F^ (D) , F^ (D} , . . . .

Combining the above results we obtain thus for u the ex-
pression

Let us in applying this expression first suppose that the
factors of f0(D~) are real and unequal, so that fQ(D] is of the

ART. 1.] ADDITIONS TO CHAPTER XVII. 179

form (D — a}(D-l}(D-c}.... Further, let us suppose that
no two of the quantities a, b} c, .... differ by an integer.

Then
whence we may assume

Thus the expression for u becomes
FJD)
F1(D)

or, since F(D] <?*=F (m) e"*,

u =

Hence, replacing e* by a;,

x*1 + Fi(a +

2) a*11 +....}

In (2) replace in like manner D by a + * and we have.
putting i for m,

+ &(« + 0 ^U(« + * -2) + ...... = o,

or, if Fi(a + i) be represented by wa+i,

«^+(/>l(«+*>a+(-1+^2(« + »>a+i_8+ ...... = 0.

Put m for a + 1", thus

um+<f>i(m} "«>-! + <!>t(m)v^+ ...... = 0.

This agrees with the law established in [there is no refer-
ence in the manuscript, but the law intended appears to be
that given in Chap. xvil. Art. 9.]

12 _ 2

180 SYMBOLICAL METHODS. [CH. XXX.

Secondly, suppose that r of the factors of f0 (D) are equal
and of the form D — a.

Then '1^ contains a term of the form

Hence the corresponding portion of u is of the form
jl + F,(D) e' + F2(D)e*e+ ...j ^(c0 + Cie+ ..... : + cr_^}
= L* + e«H-i)^ (D + a + y + e(«+WF2 (# + « + 2) +. . .1 v. . .(3)

where v stands for c0 + c,0 + c2#2 +.... + c^F"1.
Now Fi (D + a + 1) v

which on performing the differentiations becomes a polyno-
mial of the form

We see thus that (3) will assume the form of a series of terms
eae, e(a+r>e, .... each multiplied by a polynomial of the (r — l)th
degree in 6. Or arranging the terms otherwise it will con-
sist of a series of terms of the form

in which 7?0, B^ ..... Br_l are series involving e"*, e(rt+1)9,
e(a+2)fl, ..... Or lastly, changing ee to x, the portion of u in
question is of the form

+ A Gog *)

B0, 7?j, .... 5_, being polynomials in each of which the lowest
power of x is xa, and the successive powers increase by unity.

This establishes the assumption in [there is no reference
in the manuscript; probably Chap. xvil. Art. 10 is to be
supplied.]

ART. 1.] ADDITIONS TO CHAPTER XVII. 181

Thirdly, let f0 (D) contain r factors D — a^, D — at,...D — ar
in which ol5 «2, ... ar differ from each other by integers, toge-
ther with other factors.

The portion of u corresponding to the factor D — at will be

{l + Fl (D} e° + Fs (
in which

Thus Fn (D~) em0 consists of terms of the form

i being one of the numbers 1, 2, .... n. Hence Fm_i (D}
will consist of terms of the form

j being one of the numbers 1, 2, ... n. Continuing this until
i +j + k+ ... = m, we see that Fn (D} eme will ultimately con-
sist of terms of the form

i,j, k, ... receiving arbitrarily any of the values 1, 2, ... n,
and i+j+k+ ... being equal to m.

Thus the portion of u derived from A^fl will consist of
all possible terms of the form

182 SYMBOLICAL METHODS. [CH. XXX.

Let i = a, i-\-j = /3, i +j +k+ excluding the last

term = fj, ; and let the symbolical numerator which involves
only direct functions be represented by/(Z))} and we have

Af(D]

in which a, /?, .... //,, m are integers ascending by differences
not exceedin n.

[A few lines of the manuscript here are obscure, and I
venture to express in other words the idea which seems to be
involved.

Let D — a, denote one factor of f0(D), then the correspond-
ing factors in the denominator of

«&*** ..(4),

are (D - a,) (D - a, - a) ...... (D-a.-fi ......... (5).

Now if at is not greater than at} then a, + p, is less than
at + m • hence no factor in the expression (5) can be identical
with D — m — at. But if a, is greater than at, then one fac-
tor in the expression (5) may be identical with D — m — at .

Hence it follows that the denominator of the expression (4)
may contain D — m — at to the power r — 1, but not to a
higher power.]

And, since

we see that u will contain r sets of terms together of the form

A, B, (7.... being polynomials in x.

This establishes the rule in [there is no reference in the
manuscript; probably Chap. xvn. Art. 10 is to be supplied.]

ART. 2.] ADDITIONS TO CHAPTER XVII. 183

[There is no hint in the manuscript as to the position
which Article 2 was intended to occupy ; and the reasoning
does not seem fully developed.]

2. PROP. The solution of the equation

/.(£) «+/,(^) «*«+
being expressed in the form

it is not necessary to introduce new constants in interpreting
jFj(Z>), .... ; it suffices to interpret particularly if only uni-
formly and consistently.

For let
and in interpreting

let a new constant be introduced which was not in the inter-
pretation of

Xow Fm(D) em° + & (D) JF^ (D) «<***

+ fc(Z?) <*F^(D} <W+ ...... = 0,

therefore

Fm(D) e^ = - &(

hence the new constant comes from {f0(D)}~*Q, and the term
containing it must be A'P, or B'Q,...., where A'} B .... are
constants. Suppose it A'P ;

then as derived from this,

F^ (D] e<-'>« {/. (£)}-' 0 = -</>, (D
F^s (D] e^« UCD)}-1 0 = - fc (D

184 SYMBOLICAL METHODS. [CH. XXX.

Tims F De(^» = - <

The law of derivation is exactly the same as in the deri-
vation of Ft(jy) eifl from F^D) ^~1)«, ......

[Art. 3 seems intended for a reconstruction on an extended
scale of part of Chapter xvil. Art. 3.]

3. We proceed to consider more fully the theory of the
"binomial equation

eu= U.

Now the possibility of solving the equation depends upon
the nature of the symbolic function <f>(D). It is perhaps the
most general account of the present state of the theory to say
that there exist certain primary forms of this function which
render the equation solvable, and that to each of these pri-
mary forms an infinite number of the forms are reducible by
general theorems of transformation. As these theorems
admit of a statement which is independent of the form of the
function <-D we shall establish them first.

PEOP. II. The function (j> (D) in the equation

eu = U

can without otherwise changing the first member of that equa-
tion be 1st affected with any constant factor, or 2ndly con-
verted into <f>(D + a], or Srdly converted into {<j> (— D)}~1.

First. Let U=f(ee), and in the equation

i d d

let e9 = oV. Then jZ = -j^, , and the equation becomes
at/ at?

ART. 3.] ADDITIONS TO CHAPTER XTII. 185

in which D = -j^, . Thus <b(D] has been affected by a con-
ao

stant factor a.

Secondly. In the same system let M = ea*i\ Then

or

therefore v + <j> (D + a) tfv =

Here $(D] has been changed into <f>(D + a).

The result of this transformation may be conveniently ex-
pressed by the following theorem.

The equation

will be converted into
v

by the relations

u

Thirdly. In the same equation let 6 = — & ; then

d__ _d_
d6 ~ dff '

and we have

in which D=^a- Hence

do

therefore e^u + <f> (r - D} u = e^/

whence u + {<f> (r - D}}-1^ u = {<£ (r -

186 SYMBOLICAL METHODS. [CH. XXX.

In this equation let u = ere'v. Then by the last theorem,

Thus
is converted into

in which D — -=-=> , "by assuming

6 = -ff, u = ere'v.

The above transformations leave the index r in the first

ff
member unchanged. If however we assume 6 — — , whence

7/9 = a ^Iff ' we s^011^ have

^: , ?\

u + tj>(aD)ea =f(ea).

By combining this with the previous results we see
that it is possible to convert d>(D} into d>(aD+b}t and into

But the most important transformation of the function <£ (D)
is that which is established in the following proposition.

[The proposition referred to is Prop. in. of Chap. xvil.
Art. 3.]

[Article 4 was intended to follow the words " or subse-
quently, in the derivation of w" in Chap. xvil. Art. 4.]

4. It becomes therefore important to establish rules for
the treatment of the constants which in these different ways
arise.

ART. 4.] ADDITIONS TO CHAPTER XVII. 187

Xow the entire process of solution consists of three stages,
namely :

1st, the determination of F by the equation

2ndly, the solution of the transformed equation

v + Tlr(D)er°v=V,
3rdly, the determination of u by the relation

-

Let us consider these separately, supposing <f> (D] to con-
tain a single factor -=; — r whicli is made to disappear in the

generation of ^(^), so that a and ft differ by a multiple of r.
Thus the given equation is of the form

*-? .............. («)•

The transformed equation is of the form

= F,

First, suppose a—b = nr, where n is positive.
Thus

a-r] ...

SYMBOLICAL METHODS. [CH. XXX.

where Z7X is a particular value of

.

The part containing the constants will consist of terms of
the form

(D + a) . . . (D + a - nr + r} {1 - -v/r (J)} e^}'1 C<T(a'*v>
= (D + a) (D + a-r} ... (D + a-nr + r) rd

+ ty (D} e^f (D) ere + . . .1 C

Now all these terms vanish up to the o*ne containing
(«-nr)0 . therefore we have to perform the operation

C (D + a) (D + a - r) ... (D + a - nr + r) on

- r] . . . ^ (D -jr] ^'^

- r} ...^r(D -jr-r] e^^9-^ .... j,

where j = n — i — 1 ; that is, we have to perform the operation
C(D + a] (D + a-r) ... (D + a-nr+r] on

•U (D} ^ (D - r} . . . ^ (D -jr} e-^"™
+ f (D) <?ty(D}...^(D -jr} e-'0""-* + . . .j.

ART. 4.] ADDITIONS TO CHAPTER XVII. 189

Xo w ^(D}ty(D-r}...ty(D -jr) e^9

= i/r(«r - a) ty(nr - r - a) ...ty(nr-jr — a) e^"*1*

therefore we obtain

Thus this expression is the same in form for all values of
«. Therefore all the terms containing an arbitrary constant
in (7) are equivalent to only one term.

Secondly, suppose a — b = — nr.

Then u = •[(#+ 6) (D + b -r] ...(D + b-nr + r}\ v,
l- r) ... (D + b - nr + r] U.

Here there are no constants in F. But u contains n arbitrary
constants not in v, and as there is no subsequent process in
the method for destroying these or reducing them to mutual
dependence, it is necessary that the relations connecting them
should be sought by comparing the solution with that given
by the method of development in series.

NOTE. It would be better to reduce (6) to the form

before the demonstration.

[Article 5 was intended to follow Chap. xvil. Art. 7.

There is a memoir by Professor Boole on the subject of this
Article, entitled On ike Differential Equations which deter-
mine the form of the Roots of Algebraic Equations. The
memoir occupies pages 733 — 755 of the Philosophical Trans-
actions for 1864.]

190 SYMBOLICAL METHODS. [CH. XXX.

5. If we agree to regard as primary those forms of bino-
mial equations which are integrable but not through any
reduction effected by the Propositions of Art. 3, and to which
equations through the application of those propositions other
equations are reducible and so made integrable, it becomes
very important to enquire what these primary integrable
forms are. It does not appear at present possible to give a
general answer to this question, but so far as is known, such
forms if belonging to differential equations of a degree higher
than the first stand in a remarkable connexion with the
theory of algebraical equations. By the study of this theory
Mr Harley was led to the conclusion that y defined as an
implicit function of x by the algebraical equation

yn-ny + (n-l}x = 0 (8),

n being greater than 2, satisfies the binomial differential
equation

D(D-\] ... (D-n + 1)

in which ee = x. In this expression the factors of the nume-
rator are equidifferent, as of the denominator, their common

n — 1
difference being - — , but the equation is not resolvable by

72-

Propositions II. and in. into forms, the integrability which
had before been recognised.

The above result first reached by induction was confirmed
by Mr Cayley by the aid of Lagrange's theorem.

To the form (8) all algebraic equations of the third, fourth,
and fifth degrees are known to be reducible.

Mr Harley has subsequently found that y considered as a
function of x defined implicitly by the equation

y* - H?/""1 + (n - 1) x = 0
satisfies the symbolical differential equation

nn~l [(n - 1) Dp-'y - (n - 1) (nD - n - 1) [nD - 2]"-*e?y

= [n-l]n~le0

ART. 5.] ADDITIONS TO CHAPTER XVII. 191

the factorial notation according to which

[m]" = m(m - 1) (m - 2) ... (TO -n + 1)
being here adopted.

These results are implicitly involved in a more general
theorem which I shall now demonstrate.

THEOREM. If y1? y2 ..... y» are the n roots of the algebraic
equation

y- ay1 +1 = 0,

and if the m01 power of any one of these roots be represented
by u, and log a by 0, then u as a function of 6 satisfies the dif-
ferential equation

u-

.

And the complete integral of the above differential equation
trill be

Let y* = z, then the given equation may be expressed in
the form

z = b + az~*~,
in which b = — 1. Hence, by Lagrange's theorem,

the general term being

192 SYMBOLICAL METHODS. [CH. XXX.

which on effecting the differentiations and adopting the fac-
torial notation becomes

n [r]r

and this expression will be found to represent the first term
as well as the others of Lagrange's expansion provided that
we interpret the form

Further, the above general development includes the n

m

particular developments of u or y* arising from the giving

to Z>" its n particular algebraic values. In this way it repre-
sents the m^ power of each of the n roots ylt yz, yn in

succession.

Now representing the above general term by urar} we shall
have

|W(n-l)r T-'
~n l\

n[r]'

m

m + (n — 1) r l^"-1 ^+1

— n\ o

n J

Therefore, after reduction and replacing 6 by — 1,

Vm + (n - 1) r _ I""1 fr _ m _ \
ur L n J U n~ ..

[rT

It follows therefore that the complete series of which the
general term is urar will if represented by u satisfy the diffe-
rential equation

ART. 5.] ADDITIONS TO CHAPTER XVII. 193

U - ±-Z " ^ X/* €»*« = 0 ...(I).

If we integrate the equation in a series (Chap. xvii. Art. 9),
the initial terms of the value of u will be

the succeeding terms being formed from these by the law
(9). Hence, if the arbitrary constants C0Cl ..... Cn_l be so
determined as to make the above initial terms agree with
the first n terms of the Lagrangean expansion in any of its
particular forms, the succeeding terms will also agree, and
the Lagrangean expansion will thus become a particular inte-
gral of the equation (I). The aggregate of such particular
integrals, each affected by an arbitrary constant, will therefore
also be an integral of the differential equation, and will, in
fact, constitute its general integral, subject to exception only
in the case in which for a particular value of ra the integrals
y™, 2/2™> ..... y»"' cease to be independent.

For instance, if m=—l, and we reduce the equation to
the form

OT-4T*+1-^

it is seen that except when n = 2, we have

y^+

Here then the solution

.............. (10)

ceases to be general for it becomes

and virtually involves but n - 1 arbitrary constants.
If, however, we give to the integral the form

B. D. E. II. 13

194 SYMBOLICAL METHODS. [CH. XXX.

the last term of which becomes a vanishing fraction when
m = — 1, we find for the general value of u in this case

and in this way we may proceed in failing cases generally.

Lastly, it may be observed that in certain cases the differen-
tial equation (I) admits of reduction to an order lower by
unity than its own. And in particular this happens in the
failing cases above noticed. Thus, if in (I) we make m = — 1
the equation will be expressible in the form

whence, operating on both members with (D — n + 1)"1, we
have

i r.» _ i i "i

iu-~ . i_± D -i-i

n |_ n n

The general integral of this equation will be expressed by
(10) provided that a proper relation be established between G
and the constants Cj, 02, ..... Cn. If we choose to determine
C so as to give to the integral the particular form y~l, we shall
find on substituting for u its Lagrangean development making
m = — 1, b = — l, and calculating the coefficient of a""1 or
eC1"1'9 -n {kg £rg£ member of the differential,

Hence, if n be greater than 2, we have (7=0. It follows
therefore that if n be greater than 2, the equation

n n ' ''

i r« _

-
n

ART. o.] ADDITIONS TO CHAPTER XVII. 195

in which e* = a has for its general integral

« = Cj? + C&-1 . . .. + CU/V,
yj} yz, ..... y^.l being any n — 1 roots of the equation
y* - ay*'1 + 1 = 0.

It may "be useful to notice the forms which the above
results assume when & is changed into — 0, and therefore D
into — D ; see Art. 3.

It will be found that (I) becomes

u-T - : - ^-1] - -e^M = 0 ...... (Ill),

r^z.1 /?_«]• £+™

n nj \n n

of which the integral is therefore

«=C1y-+C7i3f- .....
yj} y,, ..... y» being the roots of the equation

3^-^ + 1 = 0 ................. (11);

and log a being denoted by 6 ;

while as the equivalent of (II) we have

0 ............ (IV)>

» n

of which, supposing n greater than 2, the integral is

y^y,, ..... y,-! being any n — 1 roots of the same algebraic
equation.

13—2

196 SYMBOLICAL METHODS. [en. xxx.

Mr Harley's results may readily be deduced from the
above. Thus it will be found that the equation (11) re-
duces to

tn — nt + (n — 1) x = 0
if we make

n—l «t— i

~- -- (n — I1!" T"""

rT1-- (n — 1} "r "/ 17 — ^ '

• U — I ft J. I »*/ ^« •• ^

71

Hence, making x = e6' and representing -^, by D', we have
for the transformation of (IV)

1 7

w = (n — 1) " e~»" ^.

^
Substituting and multiplying the result by e», we find

D'-

__

_ /n-n--l n-l =

( n J [^'J1'-1

which is Mr Harley's first equation.

If in (I) and (III) we make 1 -- = a, whence m = n — no.,

_ _

and at the same time change a into ab n, and y into yb ", we
shall obtain the following somewhat more general statement
of their united import.

ART. 6.] ADDITIONS TO CHAPTER XVII. 197

The differential equations

'

b • [D]

/j

(f-.H-l)

are JofA satisfied by the general integral

u =

y1} ys, .... ym are <Ae roots of the algebraic equation

provided that for the first equation a = €*, awaT for the second
a= e~*.

If n = 2, the above equations assume the forms
D

6. [The two principal papers by Mr Harley on the dif-
ferential equations exhibited on page 190 are the following :

(1) On the Theory of the Transcendental Solution of
Algebraic Equations, Quarterly Journal of Mathematics,
Vol. Y. pages 337... 360.

(2) On a certain class of Linear" Differential Equations.
Manchester Jlemoirs. Third Series. Vol. II. pages 232... 245.

198 SYMBOLICAL METHODS. [CH. XXX.

In a letter bearing date January 13, 1864, Professor Boole
pointed out to Mr Harley that his second equation might also
be deduced from the general theorem discussed in Art. 5.
Employing the above notation the deduction may be pre-
sented in the following form.

The equation (11) will reduce to

tn-nt"-:i + (n-l)x = 0,
if we make

y = (n-l}~"x~"t, a = -(n - 1)V;

tv

and for the transformation of (III) we have

1 -1 *-

e« = -(ra-l)ne% D = nD\

These substitutions being effected we arrive, after some
slight reductions, at 'the following equation,

TI" [(» - 1) 7X - m]"'1 D'u - (n - 1) \nD' -m- 1] W = 0,
which, making m = 1 and u = t, gives

nn [(n -1)D'- 1]""1 D't - (n - 1) [nDr - 2]ne*t = 0,

an equation which admits of reduction. In fact, operating on
both members with (D — I)"1, and determining the constant,
as in the former case, by the aid of the Lagrangean expansion,
we find

-n-l)\nD'- 2^efft = [n- l]"^*,
which is Mr Harley's second equation.

The references and deduction here given were to have
been added to the memoir which is cited in page 189, ac-
cording to Professor Boole's desire; but by some accident

ART. 6.] ADDITIONS TO CHAFTEE XVII. 199

they were not printed, and the omission was not discovered
until after his death.

Mr Harley has lately succeeded in obtaining the fol-
lowing extension of Professor Boole's theorem.

The differential equation

r[ <n"

a \ x -f- \ u

L ^J

[n-r d m 1n~r [r d m T ,
- - x -j- + -- 1 \- BJ ---- 1\ xru = 0,
\_ n ax n J [n ax n

is satisfied by the mlh power of any root of the equation

/-a-/-r + a = 0,
u being considered as a function of x.

From this he deduces the following ; the differential equa-

ton

[n-r d ™T~T dnr
- x -j --- \x-j-\u
[_ r ax rj [_ dxj

is satisfied by the m"1 power of any root of the equation
yn - nyn~T -f (n - 1) x = 0.

For the materials of this Article I am indebted to Mr
Harley.]

( 200 )

CHAPTER XXXI.

THE JACOBIAN THEORY OF THE LAST MULTIPLIER.

1. A SYSTEM of n differential equations of the first order
and degree containing n + 1 variables admits of n integrals
of the form

Wj, w2 , . . . un being independent functions of the original vari-
ables. When n — 1 of these integrals have been found they
enable us to eliminate n — 1 variables, with their differentials,
from the given system of equations, and so to obtain a single
final differential equation of the first order between the two
remaining variables. The final equation admits of being
made integrable by a factor, and its solution so found would
constitute the wth and last integral of the system. We pro-
pose in this Chapter to develope the theory of the above
integrating factor as established by Jacobi. The term ' prin-
ciple of the last multiplier,' which is more usually employed,
seems objectionable; for the essence of Jacobi's discovery
consisted not in demonstrating the existence or the nature of
the last integrating factor, but in the peculiar form of the
method which he gave for its determination, and in the rela-
tions which are implied in that form. The discovery may be
briefly said to consist in this; viz. that instead of forming by
means of the n — 1 known integrals the final differential
equation between two variables arid applying methods analo-
gous to those of Chap, v., to determine its integrating factor,
we construct antecedently to all integration a linear partial dif-
ferential equation of the first order, any one integral of which

ART. 1.] THE JACOBIAN THEORY. 201

will enable us to assign an integrating factor of the final differ-
ential equation, whatever the order of the previous integrations
may have been. Again, this partial differential equation de-
pending for its construction only upon the form of the system
given, we can often by examining it affirm beforehand that if
all the integrals but one of the system be in any way found,
the final integral will be deducible by quadratures. This
happens in the case of the most important of all systems of
differential equations — that of Dynamics.

Further, an ordinary differential equation of the 71th order
being reducible to a system of n differential equations of the
first order, Jacobi's theory may here also enable us to pre-
dicate the possibility of the last integration when the previous
integrations have been effected.

Beginning with a single differential equation of the first
order reduced to the form

dx _dy

Jf = T'

in which X and T are functions of the two variables x and y,
we know by Chap. v. that the integrating factor /* will be
given by the solution of the partial differential equation

dx ~dy~

the form of which should be carefully noticed-

Consider next a system of two differential equations of the
first order expressed in the general form

dx _ dy _ dz . .

T = T= ^ ........................ ( ''

X, F, and Z being functions of the three variables x, y, z, and
suppose one integral, represented by

<f>(x,y,z) = c ...................... (3),

to be known. The function </> (x, y, z), or, as we shall express

202 THE JACOBIAN THEORY [CH. XXXI.

it for brevity, <f>, will obviously satisfy the partial differential
equation

- = 0 ..... (4),

dy dz

of which indeed the given equations form the Lagrangean
auxiliary system ; see Chap. XIV.

If from the given integral we determine z as a function of
x, y and c, and substitute its value in the first of the given
differential equations, viz.

dx _dy
X~~Y'

the latter will be converted into a differential equation be-
tween x and y. But we may leave to the equation its prior
form, provided that we regard X and Y as functions of the
variables x and y, both explicitly as they appear therein, and
implicitly as they are involved in z. And this being so, the
equation (1) will become

d (pX) dz
x dz dx dy dz dy

The values of -r and - - in this equation must be found
dx dy

from the known integral (3) ; they are

dz . d(f> d<f> dz _ d<$> d<f>
dx ~ dx ' dz ' dy dy ' dz1

substituting which we have

(j> d(fiY)d<j> _Q _

dx dz dz dx dy dz dz dy

This then is the partial differential equation for determin-
ing ii. But the construction of this equation supposes </> to
be known. We propose to shew that p can be determined by
a process in which the only partial differential equation to be
solved can be constructed without the knowledge of <£.

AKT. 1.] OF THE LAST MULTIPLIER. 203

Since by actual differentiation

d_( A <fy\ _ d f A d<f>\ _dA d$ dA cfy
dx\ dzj dz\ dxj ~ dx dz dz dx '

it follows, writing /iJTfor A, that

d (fiX) d<f> d (fiX) d<f> = d / -£d$\_d_f x<ty\
dx dz dz dx dx\ dzj dz\ dxj'

Similarly

dfaY) d^ d(nY) d<f>^d f d<f>\ d f Yd*
dy dz dz dy dy \ dzj dz \ dy

Lastly, we have

,. d

Now adding the last three equations together we see that
the first member of the result vanishes by (5) : we have thus

d / d<>\ d

dz \ dx) dz V dy
The second line of the first member is equal to
d

and therefore vanishes by (4). There remains then

d f ydfa t d f vd<f>\ , d ( 7df\
-j- fuL-£ } + -j- [ft Y^- ) + -T- [fiZ -T I = 0.
dx\T dz) dy\ dzj dz\T dzj

204 THE JACOBIAN THEOKY [CH. XXXI.

Hence if we put

we have

d(MX] d(MY] d(MZ)_

', 7 r " i — VJ V /'

dx dy dz

If then by the solution of this equation a value of M dis-
tinct from 0 be found, the function -77 will be an integrating

dz

factor of that Jinal differential equation ivhich remains when z
has been eliminated from the system (2) by means of any
known integral (f> = c.

It will be observed that the equation for M is analogous in
form to the equation for p in the previous system. And this
suggests the form of the general theorem.

Thus proceeding to the case of a system of three equations

dx _ dy _ dz __ dt

we see that if

be a known integral, ^ therefore satisfying the equation

~dx* ~~dy ~dz dt~

then the system

dx _ dy _. dz
~X~~Y~~Z

will virtually involve only the variables x, y, z, since t

ART. 1.] OF THE LAST MULTIPLIER. 205

through the known integral becomes a function of x, y, z. The
equation (6) now becomes

d (MX} d(MX] dt d(MY] d(MY] dt
dx dt dx dy dt dy

d(MZ] d(MZ] dt _
dz dt dz~ '

or putting

dt d~Jf d~dr

dx dx ' dt

d(MX] <ty _d(MX) d^r d(MY) d^r d(MY] d^r
dx dt dt dx dy dt dt dy

d(MY]

~

dz dt ~ dt dz
and this is equivalent to

-

dt

dx dy dz dt

~dt dx d

and therefore becomes on rejecting the term in the second
line by (7), and putting

dx

206. THE JACOBIAN THEORY [CH. XXXI.

If from this equation a value of N distinct from 0 be ob-

N
tained, then M=-rr, and therefore

~dt

N

**~ djr cty
dt dz

This is the final multiplier, i. e. the integrating factor of
the final differential equation between x and y which remains
when z and t have been eliminated from the given system by
means of the two known integrals. In calculating p from
the above formula we must proceed as follows. The value of

-^- must be found from any given integral -^ = c ; but that of

j5- must be found from another integral from which by means

of the former one t has been eliminated. Thus the general
forms of the integrals will be

^ (a, y, z, t} = c,
$ («, y, «, c) = c'.

Lastly, the values of -—- , -^- found as above, and that of

N given by any solution (distinct from 0) of the partial dif-
ferential equation (8) having been substituted in the expres-
sion for fi, we must eliminate z and t from that expression by
means of the two known integrals. The resulting function
of ic, y, c and c' will be the integrating factor sought.

The reasoning above employed is in its nature quite inde-
pendent of the number of the equations of the original sys-
tem. The general theorem to which it leads may be thus
stated.

THEOREM. The system of n differential equations
dx d <7? d

ART. 2.] OF THE LAST MULTIPLIER. 207

"being given, if a system of n — 1 integrals

be so reduced by elimination that the variable yl shall not
appear in <£2, the variables y1? yt shall not appear in </>3, and
so on, then the integrating factor p of that final differential
equation between x and yn will be given by the formula

M

r" "" ~ »

in which J/ represents any integral distinct from 0 of the par-
tial differential equation

d(MYl] d(MYn}_

*" —

j * J

dx dyl

_

— v.

In applying this theorem the expression for p must be
freed from all the variables except x and yn by means of the
given integrals.

This is Jacobi's theorem. On account of its great importance
I propose to give another demonstration of it founded upon
the Calculus of Variations.

2. Second demonstration founded upon the Calculus of
Variations.

It will be most convenient to present the proposed system
of differential equations under the symmetrical form

dxl _ dx, _ _ dxn

-Aj JCj A,

the independent variables being a;,, ar15 ....... xn of which

Xlt Xti .... Xn are any functions. "We have thus n — 1 diffe-

rential equations, and we are to seek the integrating factor of
the differential equation which remains when by means of
Ti — 2 known integrals n — 2 of the variables with their diffe-
rentials have been eliminated.

208

THE JACOBIAN THEORY

[CH. XXXI.

Suppose P= c to be any integral of the system, then P
satisfies, and it suffices that it satisfies, the partial differential
equation

...

dx.2 ' '

" dx

Now if in place of a?l5 a?2, ....xn we introduce a new sys-
tem of independent variables M,, u2, .... un which are functions
of the former, then we shall have

dP

dx.

dP

dP

dP dP dP

C/j -j r <-J.2 ~~T~ •••• T ^-'n J 1

U , U2, •••• Un being functions of w,, wa, .... un. And by the
theory of the transformation of multiple integrals,

where

dxL''
dun

dxn

dun
dxn

The foregoing equation we may express in the form

fn dP ^ r Ui dP 7

2 Xi—r-dx.dx9...dx=2<l -fv -j— OM. au2....<
J dxt J -ti aui

Hence, representing by & an operation of differentiation
which affects only the form of P as a function of aclt x,, ... a-,

ART. 2.] OF THE LAST MULTIPLIED. 209

or of MI} MJ, ....«„, and not the independent variables them-
selves, we have

d8P

r- d8P,

J ' ~fa. l «%.-.. <w. =

and therefore integrating "by parts and equating the portions
on each side which remain under the sign of n-fold inte-
gration,

= 2 -(-£ SPrf« <fe ...u4*..

I /Z?/- V Ml J IS "

Whence again transforming the integral in the first member
J d^ H

and this beuig true quite irrespectivelj of the form of P,
we have

In this equation Jacobi's theorem is virtually contained.
For let the given equation be multiplied by any factor. Then
changing in the above X^ into MX^ and U^ into MU^ we
have

y d_(MU\

_
H"1 dx, duH '

Hence, if J/ be determined to satisfy the equation

B.D.E. II. 14

THE JACOBIAN THEORY [CH. XXXI.

^ d (MU\

Z~J- —TT~ I = U (PJ.

dui \ H J

This is wholly independent of the relations * connecting
ult ua , . . ,un with x1 , x2,....xn. Now choose the n — 2 variables
u^ w2, .... w«_2 so that w, = c,, w2 = C2> •••• w«-ij = c«-2 sna^ ^e
integrals of the given partial differential equation (a). Then
that equation transformed becomes

of which the auxiliary ordinary equation is

At the same time the equation (&) becomes
d M \ d

M

Hence -77. is the integrating factor of the preceding diffe-
rential equation between u^ and un.

Jacobi's theorem in its most general form is thus seen to
be the following

THEOREM. If the system of differential equations
^i" dx2 dv

Xl Xs Xn

be transformed by the introduction of a new system of vari-
ables ult ua, ....wn, so chosen that

f)f —^ f* O/ — T

»j — ci > uz

shall be integrals of the given system, then the final differen-
tial equation between un_l and wn shall have for its integrating

ART. 3.]

OF THE LAST MULTIPLIER.

211

factor ^, in which J/ is any function satisfying the partial
differential equation

and .ET stands for the determinant

du

dx.

The form of Jacobi's theorem obtained by the previous
demonstration may be deduced from the above by choosing
for un_l , un two of the original variables, for example xn_1 , xn ,
and transforming the integrals ult ua, ,...un_Jt so that u, shall
contain only a?2 ... xn, ua shall contain only xa...xn, and so on.

3. Jacobi has established by means of the above theorem
the very remarkable theorem that in any ordinary dynamical
problem the forces depending not upon the time but upon the
material constitution of the system, if all the integrals but
two of the dynamical equations are found, the two remaining
integrals can be found by quadratures.

1st. In a dynamical system of free points the forces act-
ing upon which depend only upon the position of the points,
we have if we represent the entire system of rectangular co-
ordinates taken in any order by x,y,z,... and the correspond-
ing resolved forces divided each by the corresponding mass
by JT, r, Z, . . . the system of equations

14—2

212 THE JACOBIAN THEORY [CH. XXXI.

or putting

*?_«/ ^-^
dt~ ' dt *'"

,7* _ ^x — dy _ dx' dy

~ x y ' ~ X~ Y '

Now as X, Y... do not contain t we may consider first the
system

dx _ dy _ dx' _ dy
~ = ~ ~~=~

and it is evident that if we can find all the integrals of this
system, t will be given by the equation

[dx
t= I — r + C,

x' having been first converted by means of the supposed in-
tegrals into a function of x.

To determine the last multiplier of the system last written
we have first the equation

d(Mx'} ^ ^ =

dx dy dx' dy

which since X, Y... do not contain x', y ... is satisfied by
M = a constant. Giving to the constant the particular value 1,
we see that if

are n — 2 integrals of the system, and if by means of these
we eliminate n — 2 of the variables and construct the differen-
tial equation between the two remaining variables, the inte-

grating factor of that equation will be jj. , in which H is the
functional determinant of w1} iia, .... un.

ART. 3.] OF THE LAST MULTIPLIER. 213

2ndly. Suppose the system subject to a material connex-
ion which establishes an equation of condition among some
or all of the co-ordinates. If we represent the co-ordinates
taken in any order and multiplied each by the square root of
the corresponding mass by x, y, ... the corresponding resolved
forces by X, Y,... and the equation of condition expressed by
means of the above modified co-ordinates by (f> = 0, the diffe-
rential equations "will be

the transformation above employed reducing all the equations
to the same type. [See the next Chapter.]

Making

dx , dy ,

= =

the system becomes

•j._dx_ dy dx' dy

"~' ~=

dy
and the Jacobian equation for H becomes

\ Jtv>\
x) [ d(My] I V

dx dy dx

I j f • • • w»

dy

Xow <f> does not contain x', y .... Let us inquire whether
it is possible to determine M also as a function of x, y ....
without x'} y .... so as to satisfy the above differential equa-
tion.

214 THE JACOBIAN THEORY [CH. XXXI.

The equation would become

<B'-T-+V'^-+... + ^( T-i'-ir + ^ jr+- 1 = 0,
dx dy \dx dx dy dy

or if we write

, d t , d _

--, -....
dx ax dy dy

and from this we must eliminate X.

Now since <j> = 0, we have by differentiating and putting
dx _ ,
dt=X>'"

M ,d£
dx 7 dy

and again differentiating

dx dt dy dt

or since

<M_

dt

md?<t> .

x -r

v

j — h -i —7 r ••••

ax ay

ART. 4.] OF THE LAST MULTIPLIER. 215

and differentiating with respect to x,

or if we make

Similarly

dy dy

Therefore

dx dx du du

t/ t/

or

\dx dx dy dy

. .... d\ d(b d\ dd>

and now eliminating T~> ~T- + -T~> -r~+ •••
0 ax ax dy dy

we obtain Q83I— J/S Q = 0,

which is satisfied by M = Q.

4 [Among Professor Boole's manuscripts I found five
pages in German, forming part of a memoir, which was pro-
bably intended for Crelle's Mathematical Journal. The
memoir was to have discussed two applications of the Calcu-

216

THE JACOBIAN THEORY

[CH. XXXI.

his of Variations; one to the Jacobian Theory of the Last
Multiplier, and the other to the Solution of Pfaff's equation

But there is only a single paragraph relating to the second
application.

The manuscript contains the same demonstration of the
Jacobian Theory of the Last Multiplier as in Art. 2 of the
present Chapter; after this demonstration some remarks occur
of which the substance will now be given.]

It is worthy of notice, that Jacobi in the 36th volume of
Crelle's Journal, deduced by the aid of the Calculus of Varia-
tions the result on which the preceding demonstration of the
Theory of the Last Multiplier depends. In fact, he shewed
that if V denotes any function of

dz

Z, -5-

dz
'" fa*

and Fbe transformed by the introduction of a new system of
independent variables w1} w2, ... un) then the following rela-
tion holds,

fdV d dV

" — ~~ """™™?*1™ ~™* ...
dxf ^ dz
1 d^r-

d rf(AF)

d cZ(AF)

ds

du

_
du*

where

A =

dx^

dxt

du, '

dun

dxn

dxn

du^

dua

ATCT. 4.] OF THE LAST MULTIPLIER. 217

Jacob! applies this result to the transformation of the ex-
pression

dx* dy* dz* '

•

But neither Jacobi himself, nor any other person, so far as I
know, has drawn attention to the application of the result
which I have given here.

[The substance of the single paragraph relating to the
second application of the Calculus of Variations will now be
given.]

Clebsch has earned the thanks of all who are interested in
the higher parts of the Theory of Differential Equations, since
he has performed the same service for Pfaff's problem as-
Jacobi did for the Theory of Partial Differential Equations of
the first order, and thereby for the equations of Dynamics.
But while I recognise the great importance of the results, I
consider it desirable to give a simpler deduction of the system
of partial differential equations therein involved, and on which
the other results depend.

( 218 )

CHAPTER XXXII.

THE DIFFERENTIAL EQUATIONS OF DYNAMICS.

[!T will be seen that this is only a fragment of the Chapter
which was to have appeared under this title.]

I do not propose in this Chapter to discuss the origin and
interpretation of the differential equations of motion or to enter
into those details of their application which are found in all or-
dinary treatises on Dynamics. But they constitute a system
analytically so remarkable from the forms in which it is
capable of being expressed, and from the general methods of
integration which emerge out of those forms, that they are
well deserving of a special attention.

Referred to rectangular co-ordinates the differential equa-
tions for the motion of a system of points free or connected
are

d(f>

+ ..

dy ay
dd>

dt

THE DIFFERENTIAL EQUATIONS OF DYNAMICS. 219

Here m is the mass at the point (x,y, z}, m that at (x',y', z),
X, Y, Z the resolved forces at (x, y, z} tending severally to
increase those co-ordinates, and so on. Lastly

are the equations of condition each of which may involve all
the co-ordinates, and X, //,... are indeterminate multipliers.

The above is usually termed the first Lagrangean form of the
differential equations. In applying it we must either elimi-
nate \, /*... from the given equations, and then by the equa-
tions of condition just so many of the co-ordinates with their
differentials, or we must retain X, /z, ... as variables so conditioned

d*x d*y

that the values of —^ , -^ ,... in the system shall satisfy iden-
dti cLti

cPx cPv
tically the differential equations involving -p-, -^,.... de-

rived from <£ = 0, i/r = 0,... viz. the equations

_

df~

The first Lagrangean system may by a slight transforma-
tion be reduced to a form in which all the equations are of
one type, viz. of the type which they would have if all the
masses were equal to unity.

For taking the first equation of the system and dividing
by »• we may express the result in the form

d*(m*x) X d<f> dty

- 7s - = " — i ~f~ A, - 1 -- {- LL -- • - . . •

from which we see that if x, y . . . had been taken to represent
the entire system of co-ordinates taken in any order and mul-
tiplied each by the square root of the corresponding mass, and
X, Y ... the corresponding resolved forces taken in the same
order and divided each by the square root of the correspond-
ing mass, the system of equations would have been

220 THE DIFFEEENTIAL EQUATIONS OF D1T 1MICS. [CH. XXXII.

A...

ax ax

all being of one type. In general investigations this form
is to be preferred.

From the first Lagrangean form another known as the
second Lagrangean, and from this again a third known as
the Hamiltonian are derived. The second Lagrangean form
is properly speaking an expression for the effect of a trans-
formation of co-ordinates in the most general sense upon the
original system, i.e. of a transformation which in place of
x, y, ... the entire system of given co-ordinates substitutes
a new system of variables £, tj, ... the expressions of which as
functions of x, y, ... are known. It is not necessary that this
new system of variables should be co-ordinates in the proper
sense of that term, determining three by three the positions
of the several masses; it suffices that they should in their en-
tirety determine and be determined by the co-ordinates given.

The second Lagrangean form may be established as
follows :

Differentiating the equations <£ = 0, i/r = 0,... with respect to
any one of the new variables £ we have

d$ dx d<j> dy _
dx d% dy d%"

d^r dx dty dy _
dx d % dg d£"

whence if we multiply the equations of the given system by

dx dy T -i i i

-f-. -£,..• and add, we have

dx d*x

*x dy d*y ~ dx ,^ dy

+ ---- * "'

( 221 )

CHAPTER XXXIII.

ON THE PEOJECTION OF A SURFACE ON A PLANE.

[THE following memoir -was found among Professor Boole's
manuscripts; a Title and Introductory Remarks were to have
been prefixed, but with this exception the memoir appears to
be finished for publication. It is sufficiently connected with
the subject of Differential Equations to find a place in the
present volume.

The memoir by Sir John Herschel to which allusion is
made is entitled, On a new Projection of the Sphere ; this was
read before the Royal Geographical Society of London on
the llth of April, 1859, and was printed as part of the Journal
of the Society, Vol. xxx. 1860, pages 100... 106. A chart of
the World on Sir John flerschel's projection has been pub-
lished by A. and C. Black of Edinburgh.

The history of the subject will be found in Chapter xxin.
of the Coup d'oeil liistorique sur la Projection des Cartes de
Geographie... Par M. D'Avezac, Paris, 1863.

For the materials of this introductory notice I am indebted
to Sir John Herschel.]

1. Let x, y, z be the rectangular co-ordinates of any point
on the given surface ; a?', y the co-ordinates of the correspond-
ing point on the plane of projection. Let the equation of the
given surface be

F(x,y,z)=0',
or, for simplicity,

222 ON THE PROJECTION [cil. XXXIII.

The condition of projection upon which Sir John Herschel's
investigations are founded, and which we shall adopt here, is
that of the similarity of corresponding infinitesimal areas on
the surface and on the plane. The object of the problem then
in general is the discovery of the mode in which x, y depend
upon 35, y, and z in accordance with the above condition; its
object in any particular case is the determination of 35', y as
functions of 35, y, z.

Regarding then x, y as ultimately functions of 35, y, z we
have

finf* f/'Y* fflf*

dx = -f- dx + -r- du + —r- dz,
dx dy ' dz

-i i dy , dy' , dy' ,

dy=ixdx+tydy+tzdz>

in which dx, dy, dz are not independent, but are connected
by the condition

dF 7 dF, dF 7

-y-oaH dy H — 7- dz = 0.

dx dy dz

Now for brevity write

dx' _ dx _ . dx _
dx dy dz

dx dy dz

dF-A ^_P dF n.
~3~~ — -"•> ~l — — 4*1 ~~T~ — ^ >
dx dy dz

then

dx' = adx + Idy + cdz (1),

dy' = a'dx + I'dy + cdz (2),

Cdz.. ..(3).

ART. 1.] OF A SURFACE ON A PLANE. 223

Now the condition of the similarity of infinitesimal cor-
responding areas may be resolved into the two following
conditions, viz.:

1st. The equality of their corresponding angles.
2ndly. The proportionality of their corresponding sides.
And these conditions we shall introduce separately.

1st. Assuming any point x, y on the plane of projection,
let x alone vary, and the infinitesimal line generated is dx,
while (since dy = 0) (2) and (3) become

a'dx 4- Vdy + c'dz = 0,
Adx + £dy+ Cdz = 0,

whence, if we write

L = Bc- CV, M= Ca - Ac, JV= Ab' - Ba>

dx dy dz
^e have ~L~~M='~N ........................... ' '*

?o that the direction cosines of the infinitesimal line on the
surface F corresponding to the line dx on the plane (x', y')
will be

(5).

In like manner, if y alone vary, we shall find for the
direction cosines of the infinitesimal line on the surface F
which corresponds to dy on the plane

'2-

where L' = Bc- Cb, M'=Ca- Ac, N' = Ab -Ba.

By the first of the conditions of similarity the angle be-
tween these lines on the surface must be a right angle since

224 ON THE PROJECTION [CH. XXXIII.

dx and dy are at right angles. Hence we have, from
(5) and (6),

Q ............................... (7).

2ndly. The ratio of the length of the element dx to the
corresponding element on the surface is

dx'

or, by (1),

a dx + bdy 4- cdz
~

and therefore by (4)

equating which to the corresponding expression for the ratio
of the length of dy to that of its projection on the surface,
we have

aL + bM+ cN a'L' + I'M' + c'N'

'* '* '

Now if we substitute for L, M, N, L', M'} N' their values,
we shall find

aL + lM + cN=A(Vc-lc}+B(c'a-cd} + C(ab-db'),

and the second members of these equations differ only in sign.
Thus (8) may be expressed in the form

A (b'c - be) + B (c'a - ca') + C (ab'-ab}\

(L'z

ART. 2.] OF A SURFACE ON A PLANE. 225

But the first factor of the first member of this equation
being the determinant of the system

adx + bdy + cdz = 0,

a'dx + Vdy + cdz = 0,

Adx + Bdy + Cdz = 0,

expresses when equated to zero the condition that if in the
system (1), (2), (3) dy vanishes dx shall also vanish; and
dx and dy being independent, this condition cannot be satis-
fied, so that (9) reduces to

= 0,

(Jj' + M- + A\-)* (LI' + M' + jv;8

whence

(10),

and this, with (7), will fully express the conditions of simi-
larity.

2. If we multiply (7) by 2 V^l, and add and subtract
the result from (10), we obtain the equivalent system

(L + L V^T)2 + (M' + M V^l)1 + (N1 + No/^i? = 0)
(L - L V^l)2 + ( Jf - M V^J )« + (j^_ Ntt}* = J * '

xr T> , T /—* dF dx dF dx

Now L +£v-l = _-^ ---

dy dz dz dy

, fdFdy dF dy

±-r--f--j--f

\dy dz. dz- dy

±'^/^T dF dx'

dy dz dz dy

B.D. E. IT. 15

226 ON THE PROJECTION

Writing then

x' + y V- 1 == u, x - y V- 1 =

we have

. /— dF du dF du

L + L v- 1 = -j- -j- - -T- j- *
rf? cte 02 fl/

[CH. XXXIII.

r— .

X/ - Ll V — 1 -- 5 — "^7 ~7~ 7 •

ay dz dz ay

In the same way

V— ^dF du _dFdu
~ dz dx dx dz '

N'

—--— —
rfic dy dy dx"1

Substituting which in the system (11) there result

fdF_ du_dF du\* /dF du_

\dy dz dz dy) \dz dx dx dz

_=
dx dy dy dx

_ dF dv _dF dv\*

\dy dz dz dy) \dz dx dx dz)

(dF dv dF dv

dF dv _ dF dv\*= Q
dx dy dv dx)

\dx dy dy

'....(12),

ART. 2.] OF A SURFACE ON A PLANE. 227

to which we may give the somewhat more convenient form

_fdFdu dFdu dFdu\*= ~

\dx dx d d dz dz) '

_(dFdv dF dv^ dF dv\*= .„-
\dx dx dy dy dz dz)

These are partial differential equations of the first order,
serving to determine u and v as functions of x, y, 2.

Bnt it is not necessary to solve the equations in their
general form. For, x, y, and z being connected by the equa-
tion of the surface, the above equations may always be so
reduced as to involve only two independent variables. As
latitude and longitude determine the position of a point on
the earth, so two co-ordinates of any given species will deter-
mine the position of .a point on the given surface, and these
co-ordinates, when fixed upon, become the independent varia-
bles of the problem.

Let s and t represent such co-ordinates, and let their ex-
pressions in terms of x, y, z give

s = & fa y, *}, t = & fa y, z),

which equations combined with that of the given surface will
reciprocally determine x, y, z as functions of s and t. Then
1st the differential coefficients of F which in the equations
(I), (II), are functions of x, y, z may be transformed into func-
tions of s and t ; 2ndly, we have

du _ du ds du dt

dx ds dx dt dx'

du _ du ds du dt

dy ds dy dt dy*

du _ du ds du dt

dz~ ds ~dz ~dt~dz'>

228 . ON THE PKOJECTION [CH. XXXIII.

and as -r- , -=-... are known functions of x, y, z, they also
dx ax

are expressible in terms of s and t. The result of these sub-
stitutions will then be to convert (I) into a partial differential
equation in which u is the dependent and s and t the inde-
pendent variables, and this equation being, like (I), of the
first order and second degree in the differential coefficients of
u, will be of the form

+

\dsj as at

For v we shall have an exactly similar equation with the
same coefficients.

The above equation is, by the solution of a quadratic,
resolvable into two equations of the form

du du __ du du _

~A ' ~^~

To these correspond the respective auxiliary equations

Q ........... (13).

If the integrals of these are
8 = ^,
respectively, then we have

Now v being determinable by an equation of the same
form as u, it follows that of the above two values of u one.
must be assigned to v, so that the solution of the problem will
be contained in the system

or in the system

ART. 3.] OF A SURFACE ON A PLANE. 229

The particular forms of the arbitrary functions (f> and ty
will depend solely upon the nature of the problem under con-
sideration.

One other point remains to be noticed. The first mem-
bers of (12) are essentially positive, being composed of
squares ; so are then the first members of (I) , (II) ; and so,
if the intermediate transformations are real, is the first
member of the equation whose coeificients are P, Q, E.
Hence the quadratic determining \t, X2 will have imaginary
roots of the form a + /3 J— 1. Ultimately therefore it will suf-
fice to integrate one equation of the system (13) and then to
deduce the solution of the other by changing J— 1 into

F~~3. Application of the above formulae, ichen the given sur-
face is an oblate spheroid, such as the earth.

Let the plane of the equator be that of projection, the
centre being the origin. Let the co-ordinates x, y pass
through the meridians of 0 and of 90° respectively, and z
through the poles. The equation of the surface will be

where a is the earth's equatorial, b its polar radius. Let also
the latitude of the point x, y, z be represented by s, the
longitude by t. We have

dF 2x dF 2 dF 2z

and substituting in (I),

p

fx du y du z du\2 _

230 ON THE PROJECTION

2

or, if we represent ™ by A2,

, , , 4 , f /<&A» /tfiA" A

(a? + / •+ AV) 4 -T- + -T- +hr

' iVfte/ Wy/ V&

[CH. XXXIII.

dx

dz

,,(15).

Now as a-, T/ are rectangular co-ordinates in the plane of
the equator, and x passes through the first meridian, we have

^ = tan t.
x

Again, representing in the annexed figure the meridian of
the point P, or (x, y, z)
touched by the straight line
QR in the same plane, we
have GMWaf+y, MP=z.
Therefore if Vaj* + yz=r, the
equation of the meridian is

rz z2

— 4- — — 1

* + ~ '

that of the tangent

rr zz _

+ 9

r, z being current rectangular co-ordinates of the tangent.
Hence

tan CQR = ^ = -=4=.

But CQE = latitude. Therefore finally

Ji2z
s = tan : .

x

(16),

and we must now transform (15) so as to make s and t the
independent variables.

AET. 3.] OF A SUEFACE ON A PLANE. 231

From the above equations combined with (14) we find

ah cost ah sin t a tang

~ * tan's '" ( '*

and substituting in (15),

f du . du du\ /tox

— (cos* j- + sm «-5- + tans^- =0 ....... (18).

V ax dy dz]

Again,

du _ du ds du dt

dx ds dx dt dx '

du _ du ds du dt

dy ds dy dt dy '

du _ du ds du dt
dz ds dz dt dz'

\r _ — z& — sin s cos s cos t

7 ~

where H= W + tanas. In like manner
ds — sin s cos s sin t

dy ah

ds _ h cos's
dz a

dt -s

ah

dt _ cos t
dy ah

S«0.

dz

232 ON THE PROJECTION [CH. XXXIII.

Hence

du JH f . du du\

r- = 3LT- — sm * cos * cos t -: sm t -7- ) .

ax an \ as at)

du JH f . du du\

-r- = ~- — sm s cos * sm t -j- + cos t -=•
ay ah \ as dt J

du JH f-,,, 2 du\
~T~ ~ j, (« cos* -7-).

Substituting these values in (18), and dividing by the com-
mon factor -272- we have on reduction

(d£}+ cos2* (1 + (hz - 1) cos2*}2 (^Y= 0,
\fitj \as /

which is resolvable into

du i — - , „ , du

-j: — J— 1 cos * {1 + (A — 1) cos *} -7- = 0,

at ' as

partial differential equations of which the integrals are in-
cluded in the common formula

_ ,/r ds /— T\

^(jcoss {1 + (A2- 1) cos2*} ± V /'
Now ' ds

— 1) cos2*}
ds , f cos * ds

~ j cos s + ^ ^ J 1

- f

"J

+ (£2 - 1 ) cos2
ds

. _
cos* ^

ART. 3.] OF A SURFACE ON A PLANE.

ds , f cos s ds f . , a* — b*\

e » • -i ? smce e ~ r-

coss J 1 — e sins \ a /

e , 1 — e sin s

f

=
J

sins

. f /I — e sm sY» /7T sM
= losr J(- I tan (- + -V.

3 [VI + e sins/ V^ 2/)

Hence

, F, (I - e sm a\t /TT *\{ / — -1

w = <i log 4 h tan - + - )> ± <V-1 ,

0 [VI + e sm s/ V4 */J J

or, changing <j> (t] into ^> (e*),
/l-esins\r

sins/ V-A 2

tan - + -

"I

,
J

— e sns\»

TT s^

+

4 2/

_
sins/

Let r and # be the polar co-ordinates of that point in the
plane of projection which corresponds to the point whose
latitude and longitude on the surface are s and t ; and let

o fl-e

\l + e sin sj
then the complete solution assumes the very simple form

^) (HI).

Of particular deductions the most interesting is that which
arises from the supposition that the parallels of latitude are
projected into circles round the pole. This requires that r

B.D.E. II. 16

234 ON THE PROJECTION [CH. XXXIII.

should be independent of t, a condition which is satisfied in
the most general manner by assuming

we then find

whence, on multiplication and division,

r =

whence, .4 and .5 being new arbitrary constants derived from
C and C'

r = ASn, 6=±nt + B.

If we observe that 6 and £ should vanish together, we have
.5 = 0, and the equation 6 = + nt shews that the surface of
the sphere will be projected into a sector of a circle, the arc
of which is to the circumference of the circle as n : 1. Thus,

if n = - , the sphere is projected upon a quadrant, and so

on.

The other equation gives

If 5 = 0 we find r = A, whence A is the distance of the
equator from the pole in the plane of projection, and if that
distance, which is arbitrary, be assumed as the unit, we have

.TT s\r fl — e sin.s
r = tan \[ - -f

42/1 \l+e sin s

for the distance from the pole of that parallel whose latitude
is s. We may give to this expression a better form by

ART. 3.]

OF A SURFACE ON A PLANE.

235

7T

assuming p = — + s, and introducing an auxiliary quantity
determined by the equation

"We have then

The following table gives the values of r for the sphere and
for the spheroid whose eccentricity is '08 (which is about that
of the earth), for each ten degrees of polar distance, for the

values n = 1, and n = - .

Polar
Distance.

n-

Sphere.

= 1

Spheroid.

n

Sphere.

1
~4

Spheroid.

10°

•0875

•OS80

•5439

•5447

20°

•1763

•1774

•6480

•6490

30°

•2679

•2694

7195

•7205

40°

•3640

•3658

•7767

•7777

50°

•4663

•4683

•8264

•8272

60°

'5774

•5792

•8717

•8724

70°

•7002

•7017

•9148

•9153

80°

•8391

•8400

•9571

•9574

90°

I'OOOO

roooo

i-oooo

i-oooo

TOO9

1-1918

1-1904

1-0448

1-0445

110°

1-4281

1-4250

1-0932

1-0926

1209

1-7321

1-7265

1-1472

1-1463

130°

2-1445

2-1357

T2101

1-2089

140°

2-747-5

2-7340

1*2875

1-2859

150"

3-7321

3-7114

1-3899

1-3880

160°

5-6713

5-6372

1-5432

1-5409

170°

11-4301

11-3581

1-8387

1-8358

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